Characters, correspondences and fields of values of finite ... · que el Teorema E tenga como corolario una caracterizaci on de los grupos que tienen un p-subgrupo de Sylow autonormalizante

Characters, correspondences andfields of values of finite groups

Carolina Vallejo Rodrıguez

supervised by

Gabriel Navarro Ortega

This dissertation is submitted for the degree

of International Doctor in Mathematics

March 17, 2016

A mis padres y hermanos

Contents

Resumen viiGuion de la tesis viii

Acknowledgements xiii

Introduction xvOutline of the thesis xvi

Chapter 1. Preliminaries on ordinary character theory of finite groups 11.1. Algebras, representations and characters 11.2. Induction and restriction of characters 51.3. Actions and characters 81.4. Basic Bπ-theory 101.5. The projective special linear group PSL2pqq 13

Chapter 2. Monomial characters and Feit numbers 192.1. Introduction 192.2. Two criteria for monomiality 212.3. Certain character correspondences 262.4. Certain monomial characters and their subnormal constituents 292.5. Feit’s Conjecture and p1-degree characters 31

Chapter 3. McKay natural correspondences of characters 373.1. Introduction 373.2. Character correspondences associated with PSL2p3

3aq 393.3. An extension theorem 433.4. The self-normalizing case 453.5. The p-decomposable Sylow normalizer case 493.6. The p-solvable case 52

Chapter 4. Preliminaries on modular character theory of finite groups 574.1. Brauer characters (as characters) 574.2. Isomorphisms of modular character triples 614.3. Fake Galois action on Brauer characters 72

Chapter 5. Coprime action and Brauer characters 795.1. Introduction 795.2. Review on character counts above Glauberman-Isaacs

correspondents 81

v

vi

5.3. More on fake Galois conjugates 835.4. Coprime action on simple groups and their direct products 875.5. The inductive Brauer-Glauberman condition 925.6. A reduction theorem 1005.7. Some examples 103

Bibliography 107

Universitat de Valencia Carolina Vallejo Rodrıguez

Resumen

Uno de los temas mas importantes en teorıa de grupos finitos es el estudiode la relacion entre los invariantes globales y locales de un grupo. Sea Gun grupo finito y p un primo. Los p-subgrupos de G son los subgruposde G cuyo orden es una potencia de p, y los subgrupos locales de G son losnormalizadores propios de p-subgrupos de G. Como paradigma de todo esto,podemos citar un teorema clasico de Frobenius, que ha inspirado muchosresultados recientes. Este teorema establece que un grupo G posee un p-complemento normal si y solo si cada uno de sus subgrupos locales tiene unp-complemento normal.

En esta tesis, nuestra atencion se centra en el estudio de caracteres degrupos finitos, tanto ordinarios (aquellos asociados a una representacion so-bre el cuerpo de los numeros complejos) como modulares (aquellos asociadosa una representacion sobre un cuerpo de caracterıstica p). Nos interesa espe-cialmente la relacion entre los caracteres de un grupo G y los caracteres desus subgrupos locales. La conjetura de McKay es el ejemplo clave del tipode problemas en el que estamos interesados. Esta conjetura es un problemacentral dentro de toda la teorıa de representaciones y de caracteres de gru-pos finitos. Si G es un grupo finito y p es un primo, la conjetura de McKayafirma que tanto G como el normalizador de un p-subgrupo de Sylow de Gtienen el mismo numero de caracteres irreducibles de grado no divisible porp. Por tanto, predice la existencia de una biyeccion entre tales conjuntos decaracteres. El capıtulo 3 de este trabajo trata de un caso particular de estaconjetura en el que no solo encontraremos una biyeccion de tipo McKay,sino que esta biyeccion sera natural.

Como hemos mencionado, nos interesa relacionar los caracteres de ungrupo G con los caracteres de sus subgrupos locales. La situacion es especial-mente atractiva cuando podemos establecer esa relacion a traves de corres-pondencias naturales. Quiza, el ejemplo mas representativo sea la correspon-dencia de Glauberman. Veamos su caso mas paradigmatico. Supongamosque un p-grupo P actua sobre un grupo K cuyo orden no es divisible por p.Entonces existe una biyeccion natural ˚ entre los caracteres irreducibles deK que son fijados por la accion de P , denotamos por IrrP pKq a este con-junto, y los caracteres irreducibles del grupo C “ CKpP q de puntos fijadospor la accion. De hecho, si χ P IrrP pKq, entonces

χC “ eχ˚ ` p∆,

vii

viii

donde e es un numero natural no divisible por p y ∆ es o bien un caracterde C o bien es cero. Por tanto, χ˚ es la unica constituyente del caracter χCque aparece con multiplicidad no divisible entre p. Vemos que χ determinacanonicamente a χ˚ y lo mismo ocurre en sentido contrario.

La palabra natural ya ha aparecido en varias ocasiones en este resumen.De hecho, volvera a aparecer en distintas situaciones a lo largo de este tra-bajo. Por tanto, conviene que aclaremos que entendemos cuando tildamosuna biyeccion de natural o canonica. Para ello, usaremos palabras de I.M. Isaacs. La siguiente cita esta extraıda (y traducida) del importantısimoartıculo de 1973 [Isa73] en el que I. M. Isaacs prueba la conjetura de McKayen el caso en que el orden de G es impar: !La palabra“natural” quiere decirque la correspondencia se construye a traves de un algoritmo y que el resul-tado es independiente de cualquier eleccion tomada al aplicar el algoritmo".

Sea χ un caracter de G. El cuerpo de valores Qpχq de χ es la menor delas extensiones de cuerpo de Q que contiene todos los valores de χ. Si nosencontramos en la situacion de tener una correspondencia de Glaubermany los caracteres χ y χ˚ se corresponden, entonces sus cuerpos de valorescoinciden Qpχq “ Qpχ˚q. Esto se debe a que la biyeccion ˚ conmuta con laaccion de automorfismos de Galois sobre caracteres. En general, esperamosque las biyecciones naturales no sean meras biyecciones entre conjuntos,es decir, que tengan propiedades adicionales. Por ejemplo, esperamos queconmuten con la accion de ciertos automorfismos de Galois y con la accion deciertos automorfismos de grupo. En este sentido, las biyecciones naturalesdeben proporcionar mas informacion que relacione la estructura global ylocal del grupo.

Guion de la tesis

Todos los grupos que consideramos en esta tesis son finitos. Los primerostres capıtulos de la tesis tratan sobre caracteres ordinarios, mientras quelos dos ultimos estan dedicados al estudio de caracteres modulares, tambienconocidos como caracteres de Brauer. Los resultados originales que contieneesta tesis aparecen en los siguientes artıculos [NV12], [Val14], [NTV14],[NV15], [SV16] y [Val16].

En el capıtulo 1 exponemos la teorıa de caracteres ordinaria basica quevamos a usar a lo largo de la tesis. Nuestra referencia para caracteres or-dinarios es [Isa76]. Tambien incluimos en este capıtulo una pequena ex-posicion de la teorıa de Bπ-characteres de Isaacs que, aunque no es ele-mental, sera usada con bastante frecuencia a lo largo de este trabajo (en loscapıtulos 2, 3 y 4). Finalmente, y para la comodidad del lector, presentamosal final de este primer capıtulo resultados bien conocidos sobre los gruposPSL2pqq, puesto que estos constituyen todo el bagaje sobre grupos simplesque necesitaremos en el capıtulo 3.


ix

En el capıtulo 2 empieza nuestro trabajo original. Un caracter lineal λde un grupo G no es mas que un homomorfismo G Ñ Cˆ. Por tanto, loscaracteres lineales son los mas faciles de comprender. Tambien, un grupo esabeliano si y solo si todos sus caracteres son lineales. Un caracter χ de G sedice monomial si esta inducido a partir de algun caracter lineal. Desde estaperspectiva, los caracteres monomiales tambien deben ser faciles de entender.Sin embargo, los grupos en los que todos los caracteres son monomiales sonmuy difıciles de entender. De hecho, varios problemas importantes sobreellos siguen abiertos [Nav10]. Ademas, no existen demasiados resultadosque garanticen que un cierto caracter es monomial. Una excepcion es unbonito resultado de R. Gow [Gow75]: un caracter racional de grado imparen un grupo resoluble es monomial. Nosotros extendemos este resultado deGow en los Teoremas A y B. Estos dos resultados son, por tanto, criteriospara asegurar la monomialidad de un caracter, y tienen que ver con loscuerpos de valores de los caracteres ası como con sus grados. Ademas, laconclusion del Teorema B nos permite construir una correspondencia naturalde tipo global/local, hecho que mostramos en el Teorema C.

Si usamos la teorıa de Bπ-caracteres de Isaacs, la conclusion del TeoremaB puede hacerse mas fuerte. Lo bueno es que esta nueva conclusion nospermite dar una nueva forma de calcular un invariable global de un grupode forma local. Este es el contenido de la seccion 2.3.

Concluimos el capıtulo 2 estudiando una conjetura de Feit [Fei80]. Si χes un caracter de un grupo G, escribimos fχ para denotar al menor naturaln de forma que Qpχq Ď Qn (donde Qn es el cuerpo ciclotomico que resultade adjuntar a Q una raız primitiva n-esima de la unidad).

Conjetura (Feit). Sea G un grupo finito y sea χ un caracter irreducible(ordinario) de G. Existe un elemento g P G de forma que el orden de g esexactamente fχ.

Esta conjetura fue probada para grupos resolubles por G. Amit y D.Chillag en [AC86]. Nosotros probamos una version global/local del teoremade Amit y Chillag en nuestro Teorema D. La prueba del Teorema D requiereel uso de la teorıa de caracteres especiales de Gajendragadkar y el uso delcaracter magico ψ que Isaacs define en su ya mencionado artıculo [Isa73].

En el capıtulo 3 estudiamos el caso autonormalizante de la conjecturade McKay. Sea p un primo y sea P un p-subgrupo de Sylow de G. En elTeorema E probamos que si NGpP q “ P y p es impar, entonces existe unabiyeccion natural entre los caracteres irreducibles de G de grado no divisiblepor p y los caracteres del grupo abeliano P P 1. Para probar el Teorema Eesta hemos necesitado la descripcion del comportamiento de los caracteresde PSL2pqq bajo la accion de automorfismos de cuerpo y un resultado generalde extension de caracteres que probamos en la seccion 3.3. Cabe mencionarque la conclusion del Teorema E es cierta sin restricciones sobre el primo psi asumimos que el grupo G es p-resoluble.


x

La correspondencia natural dada en el Teorema E esta descrita en ter-minos de la restriccion de caracteres de G a P . Como consecuencia deello, conmuta con la accion de automorfismos de Galois y con la accion decualquier automorfismo de G que estabilice a P . Estas propiedades hacenque el Teorema E tenga como corolario una caracterizacion de los gruposque tienen un p-subgrupo de Sylow autonormalizante para p impar. Discu-tiremos este corolario en la seccion 3.4.

En la seccion 3.5 estudiamos una extension del Teorema E al caso en queNGpP q “ CGpP qP . En este caso nuestra biyeccion natural solo se da entrecaracteres que pertenecen al bloque principal. En el Teorema F probamosque si NGpP q “ CGpP qP y ademas G es p-resoluble, entonces existe unabiyeccion natural de tipo McKay para G (y esta biyeccion coincide con ladada por el Teorema E cuando NGpP q “ P ).

El capıtulo 4 proporciona al lector la teorıa de caracteres modular nece-saria para desarrollar el capıtulo 5. La primera parte contiene resultadosbien conocidos sobre caracteres de Brauer, mientras que, en la segunda parte,probamos resultados mas especıficos. Procedemos de esta forma ya que lanaturaleza del capıtulo 5 es muy tecnica.

La seccion 4.1 es un compendio de resultados sobradamente conocidossobre caracteres de Brauer. Nuestra referencia en este caso es [Nav98].En la seccion 4.2, introducimos la nocion de isomorfismo central de ternasde caracteres modulares (que es analoga a la nocion de isomorfismo centralde ternas de caracteres ordinarios definida por G. Navarro y B. Spath en[NS14]) y estudiamos propiedades de las ternas de caracteres que son cen-tralmente isomorfas. En la seccion 4.3 introducimos el concepto de accionde Galois pretendida (cuando nos movemos en un terreno muy tecnico, latraduccion al castellano no es muy agradable). Estas pretendidas accionesde Galois serviran para compensar el hecho de que los automorfismos deGalois no actuan sobre los caracteres irreducibles de Brauer. En el capıtulo5 motivamos esta definicion.

En el capıtulo 5, estudiamos una version modular de la igualdad decardinales derivada de la correspondencia de Glauberman-Isaacs. Suponga-mos que un grupo A actua sobre un grupo G y que, ademas, p|G|, |A|q “1. Entonces, por la correspondencia de Glauberman-Isaacs se tiene que elnumero de caracteres irreducibles ordinarios de G que son fijados por laaccion de A coincide con el numero de caracteres irreducibles ordinarios delgrupo CGpAq de puntos fijados por la accion. K. Uno [Uno83] probo quelo mismo ocurre para caracteres de Brauer (con respecto al primo p) si elgrupo G es p-resoluble. El siguiente problema aparece en [Nav94].

Conjetura. Supongamos que un grupo A actua sobre un grupo G yque, ademas, p|G|, |A|q “ 1. Sea C “ CGpAq. Denotemos por IBrApGqal conjunto de caracteres irreducibles de Brauer de G fijados por A y por


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IBrpCq al conjunto de caracteres irreducibles de Brauer de C. Entonces

|IBrApGq| “ |IBrpCq|.

El resultado principal del capıtulo 5 es un teorema de reduccion parala conjetura anterior. Dicho de una forma mas precisa, en el Teorema Gprobamos que si todos los grupos finitos simples satisfacen lo que llamamosla condicion inductiva de Brauer-Glauberman, entonces la conjetura ante-rior se satisface para grupos cualesquiera G y A. En la ultima seccion deeste trabajo 5.7, pretendemos responder algunas preguntas naturalmenterelacionadas con el tema de este ultimo capıtulo, ası como plantear nuevascuestiones.


Acknowledgements

First of all, I would like to thank Gabriel Navarro ♥. He has been a wonderfulsupervisor, except for the fact that he totally refused to read Section 5.5 ofthe present work. Joking aside, I am extremely grateful for all that I havelearnt from him, as well as for his enormous patience and support.

I would like to thank P. H. Tiep and Britta Spath. It has been a pleasureto work with such inspiring mathematicians. Part of this work was doneat the TU Kaiserslautern, which I visited twice. I want to thank all theDepartment for their kind hospitality. A special thank you to Gunter Malle,for his always helpful comments on mathematics, and, of course, for valuablehiking lessons. The final form of this thesis owes him a lot. I also want tothank Britta and Marc for being exquisite dinner hosts in Kaiserslautern.

I am very grateful to my brother Jose A. Vallejo for his invitation tolecture (twice) at the UASLP and, therefore, for giving me the opportunityto visit Mexico and my beloved mexican family.

I would also like to thank my mathematical friends and colleagues. Someof them deserve a special mention: Enric, who has solved all my problemswith computers for more than three years; Eugenio, my best friend in Kaiser-slautern with whom I shared amazing moments at the E-center; Guillermo,who forced me into algebraic geometry and Joan, who helped me a lot whenI started my Ph.D. studies. Thank you, guys ♥.

I want to extend my gratitude to my thesis committee, for carefullyreading this manuscript.

Mi querido Manuel (aunque entienda perfectamente el ingles) mereceun agradecimiento especial por ser la persona, seguida de cerca por miamantısima mama, que mas ha aguantado toda la histeria derivada tantodel trabajo como de la ausencia de el. Por supuesto, no puedo dejar de darlas gracias a toda mi familia, especialmente a mis padres, Jose Antonio yCarolina, a los que adoro. Muchas gracias por aguantarme y por todos losregalos que espero recibir.

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Introduction

One of the main topics in the Theory of Finite Groups is the study of therelationship between the global and the local invariants of a finite group.Let G be a finite group, and let p be a prime. The p-subgroups of G arethe subgroups of G of order a power of p, and the local subgroups of G arethe proper normalizers of the p-subgroups of G. A paradigmatic example ofthis is an old theorem of Frobenius, which has inspired many recent results,that asserts that a group G has a normal p-complement if and only if everylocal subgroup has a normal p-complement.

In this thesis we focus our attention on characters: ordinary characters,associated with representations over the field of complex numbers, and mod-ular characters, associated with representations over fields of prime charac-teristic p. We are particularly interested in the relation between the char-acters of G and the characters of the local subgroups of G. A fundamentalexample of the kind of problems we are interested in is the McKay conjec-ture, which is nowadays at the center of the Representation and CharacterTheory of Finite Groups. If p is a prime and G is a finite group, then theMcKay conjecture asserts that G and the normalizer of a Sylow p-subgroupof G have the same number of ordinary irreducible characters of degreenot divisible by p (therefore predicting the existence of a bijection betweenthese two character sets). Chapter 3 of this work concerns a particular caseof this conjecture in which not only can we find a bijection but a naturalcorrespondence.

As we said, we are interested in relating the characters of G and thecharacters of the local subgroups of G. The situation is especially appealingif we can establish this relation by means of natural correspondences. Thecanonical example of the type of character correspondence that interests usis the Glauberman correspondence. If a p-group P acts on a group K oforder not divisible by p, then there is a natural bijection ˚ between the setof irreducible ordinary characters of K fixed under the action of P , denotedIrrP pKq, and the irreducible characters of the subgroup C “ CKpP q ofpoints fixed under the action. In fact, if χ P IrrP pKq, then the restrictionχC of χ to C can be written as

χC “ eχ˚ ` p∆,

xv

xvi

where p does not divide e, and ∆ is zero or a character of C. Hence we seethat χ canonically determines χ˚ and viceversa.

The word natural has appeared more than once in this introduction andwe will talk again about natural or canonical correspondences throughoutthis work. We feel it is worth mentioning what we mean by that. We shallnot give an exact definition of what it is, instead we quote I. M. Isaacs inhis landmark paper [Isa73] (in which he proved the McKay conjecture forgroups of odd order): !The word “natural” is intended to mean that analgorithm is given for constructing the correspondence and that the resultis independent of any choices made in application of the algorithm".

Let χ be an ordinary character of a group G. The field of values Qpχqof χ is the smallest field extension of Q containing all the values of χ. Inthe Glauberman correspondence situation, if χ and χ˚ are correspondents,then Qpχq “ Qpχ˚q. This is because ˚ commutes with the action of Galoisautomorphisms. In general, we expect natural correspondences of charactersto have more properties than being a mere bijection. For instance, naturalcorrespondences are expected to commute with certain Galois actions andwith the action of certain group automorphisms, and therefore they shouldprovide additional information as relating the global and the local structure.In this sense, the benefits of having natural bijections are multiple.

We come back to the Glauberman correspondence as in the third para-graph of this introduction. At first sight, it does not seem to go from global(a finite group G) to local (a local subgroup of G), but this is only superfi-cial. If a p-group P acts on a group K with p|P |, |K|q “ 1, then we form thesemidirect product G “ K ¸ P and we notice that NGpP q “ C ˆ P , whereC “ CGpP q. By using some elementary character theory, one can showthat the Glauberman correspondence implies that a natural bijection existsbetween the set irreducible ordinary characters of G of degree not divisibleby p, and the set of irreducible ordinary characters of NGpP q of degree notdivisible by p. Hence, the McKay conjecture in the case where the group Ghas a normal p-complement follows from the Glauberman correspondence.

Outline of the thesis

All the groups we will consider in this work are finite unless otherwise stated.The first three chapters of this thesis concern ordinary characters and thelast two chapters are devoted to modular characters (which are also knownas Brauer characters). The original results contained in this thesis appear in[NV12] (joint work with G. Navarro), [Val14], [NTV14] (joint work withG. Navarro and P. H. Tiep), [NV15] (joint work with G. Navarro), [SV16](joint work with B. Spath) and [Val16].

Chapter 1 is an expository chapter containing the background on or-dinary character theory needed for the rest of the work. Our reference is[Isa76]. We also include a brief exposition of Isaacs Bπ-theory since this


xvii

deep theory will be used in Chapters 2, 3 and 4 of the present work. Fi-nally, for the reader’s convenience we present some well-known results aboutgroups of type PSL2pqq that essentially constitute the background on simplegroups needed in Chapter 3.

In Chapter 2 we start our original work. A linear character λ of a groupG is just an homomorphism GÑ Cˆ. Hence, linear characters are the eas-iest to understand. A character χ is said to be monomial if there exists asubgroup U of G and a linear character λ of U such that λ induces χ. Thus,from this perspective, monomial characters should also be easy to under-stand. However, groups in which every character is monomial are actuallyvery hard to understand and still several open problems on them remainunsolved (see Section 12 of [Nav10]). Also, it is unfortunate that there arenot many conditions known for a character to be monomial. In [Gow75]R. Gow proved that an odd degree rational-valued irreducible character ofa solvable group is monomial. We extend Gow’s result in Theorems A andB. Both results are monomiality criteria which deal with fields of values anddegrees of characters. The conclusion of Theorem B allows us to constructnatural correspondences of characters of global/local type in Theorem C.

We actually found that the conclusion of Theorem B could be strength-ened by using Isaacs Bπ-theory, and this stronger conclusion leads to a newway of computing a global invariant of a group locally. This is the contentof Section 2.3.

We conclude Chapter 2 by studying a conjecture of Feit [Fei80]. For acharacter χ of a group G, we write fχ to denote the smallest integer n suchthat Qpχq Ď Qn (where Qn is the cyclotomic field obtained by adjoining aprimitive n-th root of unity to Q).

Conjecture (Feit). Let G be a finite group and let χ be an irreduciblecharacter of G. Then there exists an element g P G whose order is exactlyfχ.

This conjecture is known to hold for solvable groups by work of G. Amitand D. Chillag [AC86]. We prove a global/local version of Amit-Chillag’stheorem in Theorem D. The proof of Theorem D requires highly non-trivialresults on solvable groups: properties of Gajendragadkar special charactersand properties of the magical character ψ defined in the above-mentionedpaper of I. M. Isaacs [Isa73].

In Chapter 3 we study the self-normalizing case of the McKay conjec-ture. Let p be a prime and let P be a Sylow p-subgroup of a group G. IfNGpP q “ P and p is odd, then in Theorem E we prove that there existsa natural correspondence between the irreducible characters of G of degreenot divisible by p and the irreducible characters of the abelian group P P 1.Among other things, the proof of Theorem E requires the description of thebehavior of the character theory of PSL2pqq under the action of field auto-morphisms (given in Section (15B) of [IMN07]) and a key general extension


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theorem that we prove in Section 3.3. We mention that the conclusion ofTheorem E also holds without any restriction on p if G is assumed to bep-solvable.

The natural correspondence given by Theorem E can be entirely de-scribed in terms of the restriction of characters of G to P . Moreover, itcommutes with Galois action and the action of automorphisms of G thatstabilize P . These properties lead to an interesting consequence: a charac-terization of groups having a self-normalizing Sylow p-subgroup for odd p,that we discuss in Section 3.4.

In Section 3.5 we study an extension of Theorem E to the case whereNGpP q “ CGpP qP and p is odd, but only between characters in the re-spective principal blocks. In Theorem F, we prove that if G is p-solvableand NGpP q “ CGpP qP , then there exists a natural McKay correspondence(which in the case where NGpP q “ P is the one given by Theorem E).

Chapter 4 provides the modular character-theoretical background forChapter 5. The first part provides the basic background on Brauer charac-ters. Due to the highly technical nature of the results contained in Chapter5, in the second part of Chapter 4 we prove more specialized results onBrauer characters.

In Section 4.1 we collect well-known results on Brauer characters. Ourreference is [Nav98]. In Section 4.2 we introduce the notion of centralisomorphism of modular character triples (which is analogous to the notionof central isomorphism of ordinary character triples defined by G. Navarroand B. Spath in [NS14]) and we study its properties. In Section 4.3 weintroduce the concepts of fake Galois conjugate (modular) character triplesand fake Galois actions. Fake Galois actions try to remedy the fact that,in general, the Galois group GalpQ|G|Qq does not act on the irreducibleBrauer characters of the group G (they will be key in Chapter 5).

In Chapter 5 we study a modular version of the Glauberman-Isaacsbijection. Let A act on G with p|A|, |G|q “ 1. By the Glauberman-Isaacscorrespondence, the number of irreducible ordinary characters of G fixedunder the action of A equals the number of irreducible ordinary charactersof the subgroup CGpAq of fixed points. By work of K. Uno [Uno83], thesame holds for p-Brauer characters whenever G is a p-solvable group. Thefollowing was asked in [Nav94].

Conjecture. Suppose that a group A acts on G with p|A|, |G|q “ 1.Write C “ CGpAq. Also write IBrApGq to denote the set of irreducibleBrauer characters of G fixed under the action of A and IBrpCq to denotethe set of irreducible Brauer characters of C. Then

|IBrApGq| “ |IBrpCq|.

The main result of Chapter 5 is a reduction theorem for the above conjec-ture. More precisely, in Theorem G we prove that if every simple non-abeliangroup satisfies what we call the inductive Brauer-Glauberman condition then


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the above conjecture holds for all groups G and A. In the final Section 5.7we intend to answer (and raise) some questions related to this topic.


CHAPTER 1

Preliminaries on ordinary character theory offinite groups

Unless otherwise stated all groups considered are finite.

1.1. Algebras, representations and characters

Let F be a field. Given a group G, the group algebra F rGs of G over Fconsists of all the formal sums

ÿ

gPG

agg,

where ag P F for every g P G. The group algebra F rGs is an F -vector spacein the obvious way. The elements of G can be viewed as elements of F rGsvia the identification of g P G with the formal sum

ř

xPG axx where ag “ 1and ax “ 0 for x ‰ g. In fact, the elements of G form an F -basis of F rGs.Define the product of two elements of G in F rGs as the product in G andextends this definition of product to all F rGs by linearity. Then F rGs is anF -algebra.

An F -representation X of the group G is a homomorphism

X : GÑ GLnpF q,

where GLnpF q denotes the group of invertible matrices of size nˆn over F .The positive integer n is called the degree of the F -representation X.

Let X be an F -representation of G. We can extend X to the groupalgebra F rGs by linearity and we obtain an algebra homomorphism

F rGs Ñ MatnpF q,

where MatnpF q denotes the F -algebra of square matrices of size nˆ n overF . Conversely, any algebra homomorphism F rGs Ñ MatnpF q yields anF -representation of G by restriction to elements of G.

Two F -representations X and Y of G are said to be similar if thereexists some M P GLnpF q such that Xpgq “M´1YpgqM for every g P G.

Let X be an F -representation of G. We say that X is reducible if Xis similar to an F -representation Y of G in block upper triangular form

with at least two blocks. Note that Y has the form

ˆ

Y1 Z0 Y2

˙

and

1

2 1.1. Algebras, representations and characters

(by the product formula for matrices in block form) Y1 and Y2 are F -representations of G of degree strictly lower than the degree of X. We saythat X is irreducible if X is not reducible.

An F -character of a group G is defined as the trace function χ of an F -representation X of G (we say that X affords χ). The trace is an invariantunder similarity in MatnpF q, so the following is straightforward from thedefinitions.

Lemma 1.1. Let F be a field and let G be a group.

(a) Every F -character of G is constant on conjugacy classes of G.(b) Similar F -representations afford the same F -character.

Let χ be an F -character of G and let X : G Ñ GLnpF q be an F -representation affording χ. The number n is called the degree of χ. Noticethat the degree of χ is the degree of any F -representation affording χ. We saythat an F -character χ of G is irreducible if an F -representation affordingχ is irreducible.

If X and Y are two F -representations of G, then

Zpgq “

ˆ

Xpgq 00 Ypgq

˙

defines an F -representation of G. Let χ be the F -character afforded by Xand ψ by the F -character afforded by Y. Then, it is obvious that χ ` ψ isthe F -character afforded by Z. Thus, sums of characters are also characters.Moreover, a character χ is irreducible if χ cannot be written as the sum oftwo characters.

From now on, and unless otherwise stated, we fix F “ C. (We refer tocomplex characters just as characters or sometimes as ordinary characters.)We denote by IrrpGq the set of irreducible characters of G. The map sendingevery g P G to 1 P Cˆ is a representation of G of degree one. The characterafforded by this representation 1G is the principal character of G. Alinear character λ of G is a character of G of degree equal to 1. Inthis case, λ is a homomorphism G Ñ Cˆ. Of course, linear characters areirreducible.

Theorem 1.2. Let G be a group. The number of irreducible charactersof G is equal to the number of conjugacy classes of G.

Proof. See Corollary 2.7 of [Isa76].

We have seen that similar representations afford the same character. Inthe complex case, it is remarkable that most of the relevant informationcontained in a representation can be recovered from its trace.

Theorem 1.3. Let G be a group, two representations X and Y are sim-ilar if and only if they afford the same character.



1. Preliminaries on ordinary character theory of finite groups 3

A class function on a group G is a function ϕ : G Ñ C constant onconjugacy classes. It follows from Lemma 1.1 that every character is a classfunction. We usually denote the set of class functions by cfpGq. The setcfpGq has a structure of vector space in the natural way. It is clear that thedimension of cfpGq is equal to the number of conjugacy classes of G.

Theorem 1.4. Let G be a group. The set IrrpGq is a basis of cfpGq.Moreover, every character χ of G can be expressed as a sum of irreduciblecharacters of G.

Proof. See Theorem 2.8 of [Isa76].

By Theorem 1.4, if ψ is a character of G, then we can write

ψ “ÿ

χPIrrpGq

aχχ,

where the aχ are non-negative integers. If aχ ‰ 0, then χ is called a con-stituent of χ and aχ is the multiplicity of χ as a constituent of ψ.

Let ϕ and θ be two characters of a group G, we define the product ϕθfor each g P G as

ϕθpgq “ ϕpgqθpgq.

It can be proved that there exists a representation affording ϕθ (see Theorem4.1 and Corollary 4.2 of [Isa76]). Hence, products of characters are alsocharacters.

Let ClGpg1q, . . . ,ClGpgkq be the conjugacy classes of G (where gi P Gare conjugacy class representatives and g1 “ 1) and let χ1, . . . , χk be theirreducible characters of G (set χ1 “ 1G the principal character of G). Thek ˆ k matrix XpGq “ pχipgjqq

ki,j“1 is known as the character table of G.

The character table codifies fundamental information about the group. Thefirst column of the character table is the multiset of degrees of the irreduciblerepresentations of G.

Theorem 1.5. Let G be a group. Then

|G| “ÿ

χPIrrpGq

χp1q2.


Hence, the first column of the character table of G determines |G|. Also,as a consequence of this result, we obtain that the group G is abelian if andonly if every irreducible character of G is linear. The following is anotherfundamental relation between the degrees of irreducible characters and theorder of the group.

Theorem 1.6. Let χ P IrrpGq. Then χp1q divides |G|.



4 1.1. Algebras, representations and characters

We can define an inner product on the vector space cfpGq. Let ϕ, θ PcfpGq. We set

rϕ, θs “1

|G|

ÿ

gPG

ϕpgqθpgq P C,

where we denote by ω the complex conjugate of ω P C. It is easy to checkthat r , s satisfies the axioms of an inner product. Indeed r , s makes cfpGqinto a finite dimensional Hermitian space.

By the First Orthogonality Relation (see Corollary 2.4 of [Isa76]), ifχ, ψ P IrrpGq then

rχ, ψs “ δχψ,

where δij is the Kronecker delta symbol. In particular, IrrpGq is an orthonor-mal basis for cfpGq with respect to the inner product r , s. The SecondOrthogonality Relation (see Theorem 2.13 of [Isa76]) is a consequence ofthe first one and states that if g, h P G, then

ÿ

χPIrrpGq

χpgqχphq

is equal to |CGpgq| if g and h are conjugate, and is zero otherwise.

Let χ P IrrpGq. The map χ : G Ñ C defined by χpgq “ χpgq is a classfunction of G. By Lemma 2.2(c) of [Nav98], if V is a CrGs-module affordingχ, then V ˚ “ HomCpV,Cq is a CrGs-module affording χ. (The proof ofLemma 2.2(c) also applies for ordinary characters, we give this referencesince, unlike in [Isa76], a module affording χ is provided.) Hence χ P IrrpGqand χpgq “ χpg´1q for every g P G. We call χ the complex conjugate ofχ.

Let χ be a character of G. We define the kernel of χ as

kerpχq “ tg P G | χpgq “ χp1qu.

Lemma 1.7. Let χ be a character of G and let X be a representationaffording χ. Then

kerpχq “ kerpXq.

In particular, the kernel of χ is a normal subgroup of G.

Proof. See Lemma 2.19 of [Isa76].

We say that the character χ is faithful if kerpχq “ 1.

Let N Ÿ G and let χ be a character of G such that N Ď kerpχq. Ifwe define χpgNq “ χpgq for every g P G, then χ is a character of GN .Conversely, if χ is a character of GN then the function χpgq “ χpgNq is acharacter of G, and obviously N Ď kerpχq. In both cases, χ is irreducible ifand only if χ is irreducible (see Lemma 2.22 of [Isa76]). Usually we shallidentify χ and χ. In general, we can identify the characters of GN withthe characters of G containing N in their kernel.



It is easy to see that the linear characters of G are exactly the irre-ducible characters of G containing the commutator subgroup G1 in theirkernel. Hence we can identify the linear characters of G with the irreduciblecharacters of the abelian group GG1 and the index |G : G1| gives the num-ber of linear characters of G. The set of linear characters of G has a groupstructure given by the product of characters. In fact, the group of linearcharacters of G is isomorphic to GG1.

Let χ be a character of G. We can define a map detpχq : G Ñ Cˆ bychoosing X a representation that affords χ and setting

detpχqpgq “ detpXpgqq.

We claim that detpχq is a uniquely defined linear character of G. Actually ifX is a G-representation that affords the character χ, then detpXq : GÑ Cîs a homomorphism and thus, a linear representation. For the uniqueness,just notice that two representations afford the same character if and only ifthey are similar, in this case both have the same determinant.

Let χ be a character of G. We write opχq to denote the order of λ “detpχq as an element of the group of linear characters of G. We call opχqthe determinantal order of χ. Then opχq “ opλq “ |G : kerpλq|.

1.2. Induction and restriction of characters

Two essential features in character theory are restriction and induction ofcharacters. If ϕ is a class function of G and H ď G, then the restrictedfunction ϕH is obviously a class function of H. Also, if ϕ is a character,then ϕH is a character.

Let θ be a class function of some subgroup H of G. We define theinduced class function θG : GÑ C of θ to G by

θGpgq “1

|H|

ÿ

xPG

9θpxgx´1q

for every g P G, where 9θpyq “ θpyq if y P H and 9θpyq “ 0 otherwise. It iseasy to check that θG is a class function of G. In fact, by Corollary 5.3 of[Isa76], if θ is a character of H ď G, then the class function θG is also acharacter of G.

It is an elementary exercise to check that induction is a transitive oper-ation on characters. As a consequence, if ϕ is a character of H ď G suchthat ϕG P IrrpGq, then ϕS P IrrpSq for every H ď S ď G.

Let H ď G and let θ be a character of H. We can describe the kernel ofthe induced character θG as follows

kerpθGq “č

xPG

pkerpθqqx “ coreGpkerpθqq,

where coreGpHq is the intersection of all the G-conjugates of H, for H ď G.(See Lemma 5.11 of [Isa76]).


6 1.2. Induction and restriction of characters

Frobenius reciprocity (see Lemma 5.2 of [Isa76]) evidences that restric-tion and induction of complex characters are closely related: if ϕ is a classfunction of G and θ is a class function of H ď G, then

rϕH , θs “ rϕ, θGs.

In particular, if ϕ and θ are irreducible, then the multiplicity of θ as con-stituent of ϕH is equal to the multiplicity of ϕ as constituent of θG.

Let H ď G and let θ be a character of H. Let g P G. We can define

θgpxq “ θpxg´1q for every x P G. Then it is immediate that θg is a character

of Hg and rθ, θs “ rθg, θgs. In particular, θ is irreducible if and only if θg

is irreducible. Notice that if g P H, then θg “ θ because θ is constant onconjugacy classes of H.

Mackey’s formula relates the operations of induction and restriction ofcharacters.

Lemma 1.8 (Mackey). Let H,K ď G. Let T be a set of representatives

of the double cosets HgK. That is, G “ 9Ť

tPTHtK. Let θ be a character ofH. Then

pθGqK “ÿ

tPTpθtHtXKq

K .

In particular, if G “ HK then pθGqK “ pθHXKqK .

Proof. This is problem 5.6 of [Isa76].

Restriction and induction of characters behave well with respect to nor-mal subgroups. Let N Ÿ G. If θ P IrrpNq and g P G, then we have seen thatθg P IrrpNq. Hence conjugation defines a natural action of G on IrrpNq. LetGθ be the stabilizer of θ under this action. We call Gθ the inertia sub-group of θ in G. Note that N ď Gθ ď G. Let g P G. Then it follows fromthe definition that Ggθ “ Gθg . For H ď G, we say that θ is H-invariant ifH Ď Gθ.

Theorem 1.9 (Clifford). Let NŸ G and let χ P IrrpGq. Let θ be an irre-ducible constituent of χN and denote by θ1, . . . , θt the different G-conjugatesof θ in G with θ1 “ θ. Then

χN “ etÿ

i“1

θi,

where e “ rχN , θs.


We establish now some notation. Let N Ÿ G, θ P IrrpNq and χ P IrrpGq.If θ is such that rχN , θs ‰ 0 we say that θ lies under χ or that χ lies overθ. We write IrrpG|θq to denote the set of irreducible characters of G thatlie over θ and we sometimes write IrrpχN q to denote the set of irreduciblecharacters of N lying under χ.



By Clifford’s theorem, if N Ÿ G, χ P IrrpGq and θ P IrrpχN q, then thenumber e “ χp1qθp1q is the multiplicity of θ in χN . If Gθ “ G, this numbere is the degree of an irreducible projective representation of GN and divides|GN |. (We will talk about projective representations in Chapter 4.) Moregenerally, we have the following.

Theorem 1.10. Let N Ÿ G and let θ P IrrpNq. If χ P IrrpG|θq thene “ rχN , θs divides |G : N |.

Proof. See Corollary 11.29 of [Isa76]

The following result is fundamental and it will be often used throughoutthis work.

Theorem 1.11 (Clifford Correspondence). Let N Ÿ G and let θ be anirreducible character of N . Then

(a) If ψ P IrrpGθ|θq then ψG is irreducible.(b) The map ψ ÞÑ ψG from IrrpGθ|θq onto IrrpG|θq is a bijection.(c) Let χ “ ψG where ψ P IrrpGθ|θq. Then ψ is the unique irreducible

constituent of χGθ which lies over θ.(d) Let ψG “ χ where ψ P IrrpGθ|θq. Then rψN , θs “ rχN , θs.


If N Ÿ G, θ P IrrpNq and χ P IrrpG|θq, then by the Clifford correspon-dence, there exists a unique ψ P IrrpGθ|θq such that ψG “ χ. We say thatψ is the Clifford correspondent of θ and χ.

We discuss now some results on extension of characters. Let H ď G andϕ P IrrpHq, we say that ϕ extends to G if there exists χ P IrrpGq such thatχH “ ϕ.

Theorem 1.12 (Gallagher). Let NŸ G and let χ be an irreducible char-acter of G such that θ “ χN is irreducible. Then, the map

IrrpGNq Ñ IrrpG|θq

β ÞÑ βχ,

is a bijection.


The following results give us sufficient conditions for extending irre-ducible characters from normal subgroups. Recall that if χ is an irreduciblecharacter of a group G, we have defined the determinantal order opχq of χ.

Theorem 1.13. Let N Ÿ G and θ P IrrpNq be invariant in G. Supposethat p|G : N |, opθqθp1qq “ 1. Then there exists a unique extension χ P IrrpGqof θ such that p|G : N |, opχqq “ 1. In fact, opχq “ opθq. In particular, thisholds if p|G : N |, |N |q “ 1.



8 1.3. Actions and characters

Whenever the hypotheses of Theorem 1.13 hold, we call χ the canonical

extension of θ. We sometimes write χ “ pθ. The following are two usefulextendibility criteria (which however do not guarantee the existence of acanonical extension).

Theorem 1.14. Let N Ÿ G. Let θ P IrrpNq be G-invariant. Supposethat GN is cyclic. Then θ extends to G.


Theorem 1.15. Let NŸG. Let θ P IrrpNq be G-invariant. Suppose thatfor every prime p, the character θ extends to P for every P N P SylppGNq.Then θ extends to G.


1.3. Actions and characters

Let α : G Ñ H be a group isomorphism. Denote by gα the image of g P Gunder α. If χ is a character of G, then the map χα : H Ñ C defined by

χαphq “ χphα´1q for every h P H is a character of H. Moreover

rχα, χαs “ rχ, χs,

by straightforward computations. Consequently χα is irreducible if and onlyif χ is irreducible.

We see that AutpGq acts naturally on IrrpGq. Let ClpGq be the set ofconjugacy classes of G. Recall that we write ClGpgq to denote the conjugacyclass of the element g P G. Then AutpGq acts on ClpGq via ClGpgq

α “

ClGpgαq for every g P G and for every α P AutpGq. Notice that if α P AutpGq

and χ P IrrpGq, then χαpgαq “ χpgq for every g P G. (In the same way, if agroup A acts on a group G by automorphisms, then A acts on IrrpGq andon ClpGq.)

Theorem 1.16 (Brauer’s Lemma on the character table). Let A be agroup which acts on IrrpGq and on ClpGq. Assume that

χpgq “ χapgaq,

for all χ P IrrpGq, g P G and a P A; where ga belongs to the conjugacy classClGpgq

a. Then for each a P A, the number of irreducible characters of Gfixed by a is equal to the number of conjugacy classes of G fixed by a.


As a consequence of Brauer’s Lemma on the character table (see Lemma13.23 of [Isa76]), if a cyclic group A acts on G by automorphisms, then theactions of A on IrrpGq and on ClpGq are permutation isomorphic.

Let A andG be groups. Suppose that A acts by automorphisms onG andp|G|, |A|q “ 1 (we will say that A acts coprimely on G). We write IrrApGqto denote the subset of IrrpGq consisting of fixed points under the actionof A. Then there exists an important natural correspondence of characters



between IrrApGq and IrrpCGpAqq. When A is solvable, this correspondencewas constructed by G. Glauberman [Gla68]. If A is not a solvable group,then |A| is even by the Odd Order Theorem [FT63]. Consequently |G|is solvable of odd order. In this case, I. M. Isaacs [Isa73] gave a totallydifferent construction of the desired correspondence. T. R. Wolf [Wol78b]proved that when both constructions apply, when A is solvable and |G| isodd, they yield the same map. This is what we call the Glauberman-Isaacscorrespondence (when A is solvable we refer to the map as the Glaubermancorrespondence).

Theorem 1.17 (Glauberman-Isaacs correspondence). Suppose that Aacts coprimely on G. Let C “ CGpAq. Write IrrApGq to denote the subsetof IrrpGq consisting of fixed points under the action of A. There exists anatural correspondence

πpG,Aq : IrrApGq Ñ IrrpCq,

such that:

(a) If B Ÿ A and D “ CGpBq, then πpG,Aq “ πpD,ABq ˝ πpG,Bq.(b) If A is a p-group and we write χ˚ “ πpG,Aqpχq for χ P IrrApGq, then

χC “ eχ˚ ` p∆,

where p does not divide e and ∆ is a character of C or zero.

Proof. See Theorem 2.1 of [Wol79].

We consider another important example of group action on irreduciblecharacters: Galois action. Let n be an integer divisible by |G|. ConsiderQn the n-th cyclotomic field obtained by adjoining a primitive n-th root ofunity to Q. Then χpgq P Qn for every g P G and for every χ P IrrpGq (usingthat if M P GLmpCq with Mn “ Im, then M is similar to a diagonal matrixwhose entries are n-th roots of unity). Let E Ď C be the field of all algebraicnumbers. Then the theory described at the beginning of this chapter canbe developed over the field E instead of C, and it works exactly as well asfor C. If we write IrrEpGq to denote the irreducible E-characters of G, thenIrrEpGq “ IrrpGq (see the discussion on page 22 of [Isa76]).

Let χ P IrrpGq. If σ P GalpQnQq, then we define the function

χσ : GÑ C

as χσpgq “ χpgqσ for every g P G. Let X be an irreducible E-representationaffording χ. If σ P GalpQnQq, then by elementary field theory we canextend σ to pσ P GalpEQq. Define Xpσpgq “ pXpgqqpσ by applying pσ to eachentry of the matrix Xpgq, for every g P G. Clearly, Xpσ is an E-representationof G and

tracepXpσpgqq “ tracepXpgqqpσ “ χpgqpσ “ χpgqσ “ χσpgq.


10 1.4. Basic Bπ-theory

Hence, if χ P IrrpGq, then χσ P CharpGq. Since

rχσ, χσs “1

|G|

ÿ

gPG

χpgqσχpgqσ

“1

|G|

ÿ

gPG

χpgqσχpgqσ

“ rχ, χsσ “ 1,

we conclude that χσ P IrrpGq. Note that χpgqσ “ χpgqσ

for every g P G andfor every σ P GalpQnQq (see Lemma 20.7 of [Isa94]). Hence GalpQnQqacts on IrrpGq.

Let χ be a character of a group G, the field of values Qpχq of χ is theminimum field extension of Q containing all values of χ. Hence

Qpχq “ Qpχpgq | g P Gq.

Lemma 1.18. Let χ be an irreducible character of a group G. Let F Qbe an abelian Galois extension. Suppose that Qpχq Ď F . Then χσ is anirreducible character of G for every σ P GalpF Qq.

Proof. Let X be an E-representation of G affording χ. Write Fn “F X Qn, where n “ |G|. Since F Q is an abelian Galois extension, byTheorem 18.21 of [Isa94] FnQ is a Galois extension. Thus σFn P GalpFnQqand we can extend σFn to pσ P GalpQnQq. From the discussion precedingthis lemma we have that χpσ “ χσ is irreducible.

It is immediate to check that the Galois action on characters commuteswith the action induced by group automorphisms.

A character χ of G is said to be real if χ only takes real values (equiv-alently Qpχq Ď R). It is well-known that a group of odd order has nonon-principal real irreducible character.

Theorem 1.19 (Burnside). Let G be a group of odd order. If χ P IrrpGqis not principal, then χ ‰ χ.

Proof. See Problem 3.6 of [Isa76].

1.4. Basic Bπ-theory

Throughout this section π is a set of primes and G is a π-separable group.We write π1 to denote the complementary set of primes of π. If n is aninteger, then nπ is the greatest integer whose prime factors lie in π and suchthat nπ divides n. If n “ nπ, we say that n is a π-number, and if nπ “ 1then we say that n is a π1-number. If π consists of a single prime p, then wewrite π “ p and π1 “ p1.

The π-special characters of G where introduced by Gajendragadkarin 1979 [Gaj79] as the subset of IrrpGq consisting of characters χ withχp1q a π-number and such that for every subnormal subgroup S of G, the



determinantal order of every irreducible constituent of χS is a π-number. Ofcourse, the principal character is always a π-special character. In the wordsof Isaacs, the π-special characters of G are those characters that think thatG is a π-group. In fact, if G is a π-group, then every IrrpGq is π-special,and if G is a π1-group, the only π-special is the principal character 1G. Wecollect some properties of the π-special characters. We begin with goingdown properties.

Proposition 1.20. Let G be a π-separable group. Let M Ÿ G and letχ P IrrpGq be π-special. Then:

(a) Every irreducible constituent of χM is π-special.(b) Oπ1pGq ď kerpχq.

Proof. See Proposition 4.1 and Corollary 4.2 of [Gaj79].

The following are going up properties of the π-special characters.

Proposition 1.21. Let G be a π-separable group and let NŸG. Supposethat θ P IrrpNq is π-special.

(a) If GN is a π-group, then every χ P IrrpG|θq is π-special.(b) Assume that θ is G-invariant. If GN is a π1-group then θG has a

unique π-special irreducible constituent θ. In fact, θ extends θ.(c) θG has a π-special constituent iff θ is K-invariant for some Hall

π1-subgroup K of G.

Proof. See Propositions 4.3 and 4.5, and Corollary 4.8 of [Gaj79].

Let H be a Hall π-subgroup of G. The π-special characters of G restrictto irreducible characters of H injectively.

Proposition 1.22. Let G be a π-separable group and let H be a Hallπ-subgroup of G. Then the map χ ÞÑ χH is an injection from the set of π-special characters of G into the set of irreducible characters of H. Moreover,if χ is a π-special character of G, then Qpχq Ď Q|G|π .

Proof. See Propositions 6.1 and 6.3 of [Gaj79].

The following feature about special characters is particularly surprising.Although it is not common that a product of irreducible characters remainsirreducible, for special characters the following is true.

Proposition 1.23. Let G be π-separable. Let α be a π-special characterof G and β a π1-special character of G. Then αβ is irreducible. Moreover,if αβ “ α1β1 for some π-special α1 and some π1-special β1, then α “ α1 andβ “ β1.

Proof. This is Proposition 7.2 of [Gaj79].

Gajendragadkar’s π-special characters are the foundation of Isaacs’ Bπ-theory. In the important paper [Isa84], Isaacs defined a canonical set BπpGqof IrrpGq whose elements are called Bπ-characters of G. The definition of


12 1.4. Basic Bπ-theory

BπpGq is rather involved and we only give a few details below (for furtherdetails see Sections 4 and 5 of [Isa84]). The Bπ-characters can be seen as ageneralization of the π-special characters. In fact, the π-special charactersof G are exactly those elements in BπpGq of π-degree (see Lemma 5.4 of[Isa84]) and every Bπ-character is induced from some π-special character.In particular, 1G P BπpGq.

Perhaps, the most important application of this theory is the fact thatBp1-characters constitute a canonical lift of the so called p-Brauer characters(on which we will talk in Chapters 4 and 5) in p-solvable groups.

We say that χ P IrrpGq is π-factorable if there exist a π-special α PIrrpGq and a π1-special β P IrrpGq such that χ “ αβ. Suppose that Gis π-separable. For every χ P IrrpGq, there exists a particular pair pW,γqwhere W ď G, γ P IrrpW q is π-factorable and γG “ χ. This pair pW,γqis determined up to G-conjugacy. Any such pair is called a nucleus forχ, and the character in the pair is called a nucleus character for χ. Letχ P IrrpGq. Then χ is a Bπ-character if some nucleus character for χ isπ-special. Notice that if χ is a Bπ-character, then every nucleus characteris π-special, by the G-conjugacy property of the nuclei. We write BπpGq todenote the set of Bπ-characters of G.

The following properties of Bπ-characters remind of those of π-specialcharacters.

Theorem 1.24. Let G be a π-separable group and let χ P BπpGq. Then:

(a) For any N Ÿ G, the irreducible constituents of χN lie in BπpNq,(b) Oπ1pGq ď kerpχq, and(c) Qpχq Ď Q|π|.

Proof. See Corollaries 5.3, 7.5 and 12.1 of [Isa84].

The set BπpGq is closed under group automorphisms and Galois action.

Theorem 1.25. Let G be π-separable. Let χ P BπpGq, α P AutpGq andσ P GalpQ|G|Qq. Then χα and χσ lie in BπpGq.

Proof. Follows from the definition of Bπ-characters in [Isa84].

We will also need some more recent results concerning the charactertheory of π-separable groups.

Theorem 1.26. Let G be a π-separable group. Let ψ P BπpGq andsuppose that pW,γq is a nucleus for ψ. Then, the map α ÞÑ pαγqG is aninjection from the set of π1-special characters of W into IrrpGq. Moreover,let pWi, γiq be nuclei for ψi P BπpGq and let αi P IrrpWiq be π1-special, fori “ 1, 2. If pα1γ1q

G “ pα2γ2qG, then the pairs pWi, γiq for i “ 1, 2 are

G-conjugate.

Proof. See Theorems 9.1 and 9.2 of [Nav97].

In [IN01] the authors refer to the irreducible characters pαγqG given byTheorem 1.26 as the satellites of ψ P BπpGq. They note that the satellites



of ψ P BπpGq do only depend on ψ and that ψ is a satellite of itself. In fact,the second part of Theorem 1.26 implies that the sets of satellites of distinctmembers of BπpGq are disjoint.

Satellites will be useful for us mainly because of the following result.

Theorem 1.27. Let G be a π-separable group. Every χ P IrrpGq ofπ1-degree is a satellite of a unique ψ P BπpGq of π1-degree.

Proof. See Theorem 3.6 of [IN01].

Therefore, in a π-separable group G, for every χ P IrrpGq of π1-degree,Theorem 1.27 guarantees the existence of a pair pW,γq, where W ď G andγ P IrrpW q is π-special, and a π1-special α P IrrpW q such that χ “ pαγqG.

1.5. The projective special linear group PSL2pqq

In Chapter 3 we will need to study certain character correspondences ingroups PSL2p3

3aq for a ě 1. We will need to understand the action of fieldautomorphisms on characters. We include this section in order to makethis work as self-contained as possible. Throughout this section p is an oddprime. Let q “ pf . We write G “ GL2pqq, H “ SL2pqq, Z “ ZpHq andS “ PSL2pqq “ HZ. We write F “ Fq to denote the Galois field of qelements, and let α be a generator of the cyclic multiplicative group Fˆ.Then Fp is the prime field of F .

1.5.1. Automorphisms of PSL2pqq. Let δ “

ˆ

α 00 1

˙

P G. Then δ

is an element of order q ´ 1 that acts on H asˆ

a bc d

˙δ

“

ˆ

a α´1bαc d

˙

,

for every

ˆ

a bc d

˙

P H. We notice that

δ2 “

ˆ

α 00 α

˙ˆ

α 00 α´1

˙

P ZpGqH,

and thus δ2 acts on H as an inner automorphism (We remark that in thecase where q is even, δ actually acts as an inner automorphism because everynon-zero element of F is a square). We have that δ stabilizes Z. Hence δinduces an automorphism of S of order q ´ 1, which we denote again by δ,via

pxZqδ “ xδZ,

for every x P H. The automorphisms in xδy ď AutpSq are called diagonalautomorphisms. We have δ2 induces an inner automorphism of S. So ifwe embed S Ÿ AutpSq, then we have that xδy X S “ xδ2y.

Let ϕ be the Frobenius automorphism ϕ : F Ñ F of F given by ϕpaq “ap P F for every a P F . We have that ϕ P GalpF Fpq generates the full Galois


14 1.5. The projective special linear group PSL2pqq

group GalpF Fpq. The Frobenius automorphism ϕ induces an automorphismof H by applying ϕ to each entry of a matrix x P H (also in G). Then ϕstabilizes Z, and hence, ϕ defines an automorphism of S, which we denoteagain by ϕ, via

pxZqϕ “ xϕZ,

for every x P G. If we embed S Ÿ AutpSq, we have that xϕy X S “ 1. Tosee this, suppose that some ϕj ‰ 1 acts on H as the inner automorphism

associated to x “

ˆ

a bc d

˙

P H. In particular, for every e P Fˆ, we would

haveˆ

a bc d

˙

˜

epj

0

0 pepjq´1

¸

“

ˆ

e 00 e´1

˙ˆ

a bc d

˙

.

This yields ϕjpeq “ e´1 for every e P Fˆ and a “ 0 “ d. However, if we

compute the action of ϕj on elements of type

ˆ

1 g0 1

˙

P H for g P F , we

get a contradiction. The automorphisms in xϕy are called field automor-phisms.

The automorphisms δϕ and ϕδ of H differ by an inner automorphism ofH, therefore the same holds in Aut(S). It is straightforward to check that

for every x “

ˆ

a bc d

˙

P H

xϕ´1δ´1ϕδ “

ˆ

a pα´1qαpbαpα´1qpc d

˙

.

Since pα´1qαp “ αp´1 P xα2y, we have that ϕ´1δ´1ϕδ P xδ2y ď S. Writeδ “ δS P AutpSqS and ϕ “ ϕS P AutpSqS. Thus, δ and ϕ commute inthe outer automorphism group OutpSq “ AutpSqS of S.

We can embed PGL2pqq into AutpSq. Let ι : GÑ AutpSq be defined byx P G sends yZ to yxZ for every y P H. Then ι is a homomorphism withkernel ZpGq. We identify PGL2pqq with ιpGq, so that PGL2pqq ď AutpSqand S is a subgroup of PGL2pqq of index 2. Since xδy ď PGL2pqq is not

contained in S we have that PGL2pqq “ Sxδy. Also S X xδy “ xδ2y. In fact,

there exists an automorphism γ of S such that

PGL2pqq “ S ¸ xγy.

If q ” 3 mod 4, then let

γ “ ι

ˆ

´1 00 1

˙

P AutpSq.

If q ” 1 mod 4, then ´1 is a square in F . Let ε P F be such that ε2 “ ´1.Then let

γ “ ι

ˆ

0 ´1ε 0

˙

P AutpSq.



It is well-known that every automorphism of S is a product of an innerautomorphism, a diagonal automorphism and a field automorphism (see forinstance Theorem 12.5.1 of [Car93]). Thus

AutpSq “ PGL2pqq ¸ xϕy “ pS ¸ xγyq ¸ xϕy.

In particular |AutpSq| “ fqpq2 ´ 1q. We have noticed that rδ, ϕs “ 1 inOutpSq. In fact,

OutpSq “ xδy ˆ xϕy.

Suppose that some δϕj acts on H as an inner automorphism of H. By

computing the action of δϕj on elements of the form

ˆ

a 00 a´1

˙

P H andˆ

1 b0 1

˙

P H, we get a contradiction.

1.5.2. Conjugacy classes and irreducible characters of PSL2pqq.Let E be a quadratic extension of F , so that |E| “ q2. Let G “ GalpEF qand µ be the nontrivial element of G. It is easy to show that µ is exactlythe automorphism of E taking every z P E to zq P E. Since q is odd, wecan fix ε P Fˆ a non-square so that E “ F r

?εs is a quadratic extension of

F . Notice that any z P E has the form z “ a` b?ε. We write z “ a´ b

?ε,

so that µpzq “ z. Now, every quadratic polynomial rrXs P F rXs factors inErXs. Thus, every x P H has eigenvalues in E.

Let x P H. Then there are three possibilities for the eigenvalues λ, λ´1

of the matrix x:

(a) λ “ λ´1 P t1,´1u,(b) λ ‰ λ´1 in Fˆ and(c) λ ‰ λ´1 not in Fˆ.

Case (a): If λ “ λ´1 P t1,´1u then x is H-conjugate to one of thefollowing

I,Í, u “

ˆ

1 10 1

˙

,ú, u1 “

ˆ

1 ε0 1

˙

,ú1.

Indeed, suppose x ‰ λI and let

ˆ

ab

˙

be an eigenvector of x associated to

the eigenvalue λ. We first assume a ‰ 0. Let y “

ˆ

a 0b a´1

˙

P H, we have

that xyˆ

10

˙

“ λ

ˆ

10

˙

. By matrix calculation, it follows xy “

ˆ

1 c0 1

˙

or xy “

ˆ

´1 c0 ´1

˙

for some c P Fˆ. Finally, conjugating by an element

of the form

ˆ

d 00 d´1

˙

for some d P Fˆ, we get that x is conjugate to ˘u

if c is a square and x is conjugate to ˘u1 if c is a non-square. In case a “ 0,we have that b ‰ 0. Then an analogous argument with b playing the role ofa shows that x is conjugate to ˘u or ˘u1.



Case (b): If λ ‰ λ´1 lie in Fˆ, then the matrix dpλq “

ˆ

λ 00 λ´1

˙

defines a conjugacy class. We have that dpλq is conjugate to dpλ´1q anddpλq is not conjugate to other matrix of this form since the eigenvalues aresimilarity invariants. There are 1

2pq ´ 3q such classes, the same as pairs

tλ, λ´1u Ď Fˆ.

Case (c): If λ ‰ λ´1 do not lie in Fˆ. Write T “ tz P Eˆ | zz “ 1u.Then λ P T . Notice that T “ tc ` d

?ε : c, d P F and c2 ´ d2ε “ 1u. Since

the norm map Eˆ Ñ Fˆ is surjective, we have that T is a cyclic group oforder q ` 1. An element c ` d

?ε acts on E, which is a vector space over

F , and can be represented as the matrix

ˆ

c dεd c

˙

P H, with respect to

the F -basis t1,?εu. Clearly T is isomorphic to the group of matrices of this

form, so we write T ď H under this identification. Let t P T ´ Z. Then tdefines a non-central conjugacy class in H. We have that t is conjugate tot´1. Moreover, if t1 R tt, t´1u, then the eigenvalues of t and t1 are distinct, sothat the matrices t and t1 are not conjugate in GL2pq

2q. In particular, theyare not conjugate in H. Thus, each pair tλ, λ´1u Ď T defines a conjugacyclass, and there are 1

2pq ´ 1q such classes.

Write d “ dpαq “

ˆ

α 00 α´1

˙

. Let t be a generator of the subgroup

T ď H. By Theorem 38.1 of [Dor71], the set

tI,Í, u,ú, u1,ú1, dl, tmu,

where 1 ď l ď 12pq ´ 3q and 1 ď m ď 1

2pq ´ 1q, is a complete set of

representatives of the conjugacy classes of H. Write e “ p´1qq´1

2 . Let ρbe a primitive pq ´ 1q-th root of unity and let σ be a pq ` 1q-th primitiveroot of unity. By Theorem 38.1 of [Dor71], the character table of H is thefollowing:

Class: I Í u u1 dl tm

1H 1 1 1 1 1 1

StH q q 0 0 1 ´1

ξ112pq ` 1q 1

2epq ` 1q 1

2p1`

?eqq 1

2p1´

?eqq p´1ql 0

ξ212pq ` 1q 1

2epq ` 1q 1

2p1´

?eqq 1

2p1`

?eqq p´1ql 0

η112pq ´ 1q ´ 1

2epq ` 1q 1

2p´1`

?eqq 1

2p´1´

?eqq 0 p´1qm`1

η212pq ´ 1q ´ 1

2epq ` 1q 1

2p´1´

?eqq 1

2p´1`

?eqq 0 p´1qm`1

χi q ` 1 p´1qipq ` 1q 1 1 ρil ` ρil 0

θj q ´ 1 p´1qjpq ´ 1q ´1 ´1 0 ´pσjm ` σ´jmq

where 1 ď i, l ď 12pq ´ 3q and 1 ď j,m ď 1

2pq ´ 1q. As in [Dor71], thecolumns for the classes ú, ú1 are omitted. These values can be obtained

from the relations χpúq “ χpÍqχpIq χpuq and χpú1q “ χpÍq

χpIq χpu1q, for every



χ P IrrpHq. Just notice that if χ P IrrpHq and X is a representation affording

χ, then XpÍq “ λI, where λ P t˘1u and λ “ χpÍqχpIq .

Let D “ xdy ď H. We have that D is isomorphic to Fˆ. Let U be thesubgroup of H consisting of upper unitriangular matrices. It is easy to see

that NHpUq “ t

ˆ

a b0 a´1

˙

| a P Fˆ, b P F u and NHpUqU – D. Every

ξ P IrrpDq can be viewed as a character of NHpUq containing U in its kernel.Under this identification, the character ξH is called the principal seriescharacter of H associated to ξ (see Section 2.3 of [Bon11] for properties ofthe principal series characters). The character ξH is irreducible if and onlyif ξ is non-real. Moreover ξ and ξ yield the same principal series character.The characters χi in the above character table are principal series charactersassociated to some non-real ξ P IrrpDq. We write χi “ χξ, if χi comes from

the pair tξ, ξu. If ξ P IrrpDq is real-valued, then ξH decomposes as thesum of two irreducible characters. By Mackey’s Lemma 1.8, we have thatp1NHpDqq

H “ 1H ` StH , where StH is the Steinberg character of H. Let ξ0

be the only non-principal real-valued character of D, namely ξ0pdlq “ p´1ql.

Then pξ0qH “ ξ1 ` ξ2. We write ξ10 “ ξ1 and ξ20 “ ξ2.

Recall T “ xty ď H is cyclic of order q` 1. Every η P IrrpT q has associ-ated a virtual character πη of H (the description of πη is more complicatedthan for principal series characters, see Section 4.3 of [Bon11] for propertiesof πη). In fact, π1T “ StH ´ 1H and πη is actually a character wheneverη is non-principal. The irreducible constituents of πη, for η non-principal,are cuspidal characters of H. If η P IrrpT q is non-real, then πη “ πη isirreducible. The characters θj in the above character table are associatedto pairs tη, ηu Ď IrrpT q. We write θη “ θj if θ is associated to the pairtη, ηu. The characters η1 and η2 are the irreducible constituents of πη0 thecharacter associated to the unique real-valued irreducible character η0 ‰ 1Tof T , namely η0pt

mq “ p´1qm. We write θ10 “ θ1 and θ20 “ θ2.

Recall that e “ 1 if q´12 is even and e “ ´1 otherwise. Hence, ξ0pÍq “

e “ ´η0pÍq. We write eξ “ ξpÍq for ξ P IrrpDq and eη “ ηp´1q forη P IrrpT q. We re-write the character table of H in this new notation:

Class: I Í u u1 dl tm

1H 1 1 1 1 1 1

StH q q 0 0 1 ´1

ξ1012pq ` 1q e

2pq ` 1q 1

2p1`

?eqq 1

2p1´

?eqq ξ0palq 0

ξ2012pq ` 1q e

2pq ` 1q 1

2p1´

?eqq 1

2p1`

?eqq ξ0palq 0

η1012pq ´ 1q ´ e

2pq ` 1q 1

2p´1`

?eqq 1

2p´1´

?eqq 0 ´η0ptmq

η2012pq ´ 1q ´ e

2pq ` 1q 1

2p´1´

?eqq 1

2p´1`

?eqq 0 ´η0ptmq

χξ q ` 1 pq ` 1qeξ 1 1 ξpalq ` ξpalq 0

θη q ´ 1 pq ´ 1qeη ´1 ´1 0 ´ηptmq ´ ηptmq

As before 1 ď i, l ď 12pq´3q and 1 ď j,m ď 1

2pq´1q and the columns for theclasses ú, ú1 are omitted. With this notation χξpúq “ χξpú

1q “ eξ for



every non-real ξ P IrrpAq, and θηpúq “ θηpú1q “ éη for every non-real

η P IrrpT q.

The irreducible characters of S are those of H which contain Z in theirkernel, so we can calculate the character table of S from the character tableof H. In Chapter 3, we will be interested in the case where q “ 33a for somea ě 1, hence in the case where q ” 3 mod 4. Assume q ” 3 mod 4, so that12pq´1q is odd and thus e “ ´1. Since ´1 is not a square of F (because the

subgroup of squares of Fˆ has order 12pq ´ 1q), we can fix ε “ ´1. Hence,

the subgroup T of H consists of matrices of the form

ˆ

c ´dd c

˙

for c, d P F

with c2 ` d2 “ 1. Set w “

ˆ

0 ´11 0

˙

P T . Provided that xZ “ ´xZ for

every x P H, it is easy to check that a complete set of representatives of theconjugacy classes of S is

tIZ, uZ, u1Z,wZ, dlZ, tmZu,

where 1 ď l ď 14pq ´ 3q and 1 ď m ď 1

4pq ´ 3q. Notice that w “ t14pq`1q.

If 14pq ´ 3q ď j ď 1

2pq ´ 1q, then ´tj is conjugate to ´t´j “ tq`1

2´j . Thus

tjZ “ ´tjZ defines the same class as tmZ, where m “q`1

2 ´ j. Also

notice that ω “ t14pq`1q has order two in S. The character table of S is the

following:

Class: IZ uZ u1Z dlZ tmZ

1S 1 1 1 1 1

StS q 0 0 1 ´1

η10 12pq ´ 1q 12p´1` i?qq 12p´1´ i

?qq 0 ´η0ptmq

η20 12pq ´ 1q 12p´1´ i?qq 12p´1` i

?qq 0 ´η0ptmq

χξ q ` 1 1 1 ξpalq ` ξpalq 0

θη q ´ 1 ´1 ´1 0 ´ηptmq ´ ηptmq

Here 1 ď l ď 14pq ´ 3q and 1 ď m ď 1

4pq ` 1q.Consider ϕ P AutpSq. We see that ϕ fixes the classes defined by IZ,

uZ, u1Z and wZ, and permutes the classes of type dlZ and the classes oftype tmZ. In particular, ϕ fixes the trivial character 1G and the Steinbergcharacter StG. Also notice that pη10q

ϕ has degree 12pq ´ 1q and takes the

same values as η10 on uZ and u1Z. We conclude that pη10qϕ “ η10. Similarly

pη20qϕ “ η20 . Furthermore, for every non-real ξ P IrrpDq and every non-real

η P IrrpT qχϕξ “ χξϕ and θϕη “ θηϕ .

Hence, we can characterize the irreducible characters of S fixed by ϕ interms of the irreducible characters of D and T fixed by ϕ.


CHAPTER 2

Monomial characters and Feit numbers

2.1. Introduction

Let G be a finite group. There are few results guaranteeing that a singleirreducible character χ P IrrpGq is monomial. Recall that χ is monomial ifthere exist a subgroup U ď G and a linear character λ P IrrpUq such thatλG “ χ. For instance, if G is a supersolvable group, then every irreduciblecharacter of G is monomial (see Theorem 6.22 of [Isa76]). However, thisresult depends more on the structure of the group rather than on proper-ties of the characters themselves. An exception is a lovely result by Gow[Gow75] from 1975: an odd degree rational-valued irreducible character ofa solvable group is monomial. The first aim in this chapter is to generalizeGow’s result.

It is convenient now to define the Feit number of a character. If χ PIrrpGq, then we have already mentioned that Qpχq Ď Q|G|, so there is asmallest integer fχ such that Qpχq Ď Qfχ . The number fχ is called the Feitnumber of χ.

Our first original result, which generalizes Gow’s result, has been pub-lished by the author in [Val14].

Theorem A. Let G be a solvable group. Let χ P IrrpGq. If χp1q isodd and pχp1q, fχq “ 1, then there exists a subgroup U ď G and a linearcharacter λ of U such that λG “ χ. Moreover, if µ is a linear character ofsome subgroup W ď G such that µG “ χ, then there exists some g P G suchthat W “ Ug and µ “ λg.

Notice that we not only prove the monomiality of χ, but also uniquenessin the induction. The conditions χp1q odd and G solvable are necessaryas they are for Gow’s original result: SL2p3q has a rational-valued non-monomial character of degree 2, and A6 has a rational-valued non-monomialcharacter of degree 5. The new condition pχp1q, fχq “ 1 is also necessary:Let G be SmallGroupp108, 15q. We checked with GAP that every irreduciblecharacter χ of degree 3 is not monomial and fχ “ 3.

In particular, Theorem A guarantees that an odd degree irreduciblecharacter of a solvable group with values in some Q2a for a ě 0 is monomial.In Theorem B below we generalize this latter statement for odd primes,although an oddness condition is still necessary. Let p be a prime. We recallthat Irrp1pGq denotes the subset of irreducible characters of G that havedegree not divisible by p.

19

20 2.1. Introduction

Theorem B. Let G be a p-solvable group for some prime p. Let P P

SylppGq. Assume that NGpP qP has odd order. If χ P Irrp1pGq takes valuesin Qpa for some a ě 0, then there exist a subgroup U and a linear characterλ of U with Qpλq Ď Qpa such that χ “ λG. Also, if µ is a linear characterof some subgroup W ď G such that µG “ χ, then W “ V g and µ “ λg forsome g P G. In particular Qpµq Ď Qpa.

Theorem B has been published in a joint work of the author with G.Navarro [NV12]. Notice that the condition |NGpP qP | odd is superfluousif p “ 2. Unfortunately, this oddness condition is necessary in general: ifp “ 3, then SL2p3q has a rational-valued non-monomial character of degree2. The p-solvability condition is also necessary. For instance, let G “ SL3p3qand p “ 3. In this case, |NGpP q : P | “ 3 for a Sylow p-subgroup P of G.The group G has a rational-valued character of degree 12 which cannot beinduced from any proper subgroup of G.

By using non-trivial Isaacs Bπ-theory the conclusion of Theorem B canbe strengthened: such a χ is a Bp-character. We prove this fact in Section2.4. It does not seem easy at all how to control the behavior of the normalconstituents of χ without using this deep theory. We also use Bπ to providean alternative prove of Theorem A above.

Let G be a finite group, let p be a prime and let a ě 0. How manyp1-degree irreducible characters does G have with field of values containedin Qpa? It does not seem easy at all how to answer this question in general.However, if G is p-solvable and NGpP qP has odd order, then this numbercan be computed locally. We write XpapGq “ tχ P Irrp1pGq | Qpχq ĎQpau. We prove that there exists a canonical bijection from XpapGq ontoXpapNGpP qq. Theorem C below also appears in [NV12].

Theorem C. Let G be a p-solvable group and let P P SylppGq. WriteN “ NGpP q and assume that NP has odd order. Define a map

Ω : XpapGq Ñ XpapNq

in the following way: If χ P XpapGq, choose a pair pU, λq where P ď U ď Gand λ P IrrpUq is linear with Qpλq Ď Qpa such that λG “ χ, then setΩpχq “ pλUXN q

N . Then Ω is a well-defined canonical bijection.

At the end of this chapter, we will come back to Feit numbers. The Feitnumber fχ is a classic invariant in character theory that has been studiedby Burnside, Blichfeldt and Brauer, among others. But it was W. Feit whofollowing work of Blichfeldt made an astonishing conjecture that remainsopen until today (see for instance [Fei80]).

Conjecture (Feit). Let G be a finite group and let χ P IrrpGq. Thenthere exists an element g P G whose order is exactly fχ.

If G is an abelian group, then Feit’s Conjecture holds for every χ P IrrpGqsince G is isomorphic to IrrpGq. In [AC86], G. Amit and D. Chillag provedFeit’s Conjecture for solvable groups.


2. Monomial characters and Feit numbers 21

We prove a global/local variation (with respect to a prime p) of theAmit-Chillag theorem (for odd-degree characters of p1-degree).

Theorem D. Let p be a prime and let G be a solvable group. Letχ P IrrpGq of degree not divisible by p, and let P P SylppGq. If χp1q is odd,then there exists g P NGpP qP

1 such that opgq “ fχ. In particular, the Feitnumber fχ divides |NGpP q : P 1|.

Theorem D appears in [Val16]. It is unfortunate that we really needto assume that χp1q is odd, as G “ GL2p3q shows us: if χ P IrrpGq is non-rational of degree 2, then fχ “ 8; but the normalizer of a Sylow 3-subgroupof G has exponent 6. Also, Theorem A is not true outside solvable groups,as shown by G “ A5, p “ 2, and any χ P IrrpGq of degree 3 (which hasfχ “ 5).

This chapter is structured in the following way: In Section 2.2 we proveour two monomiality criteria, namely Theorem A and Theorem B. In Section2.3 we study character correspondences and we prove Theorem C. In Section2.4 we strengthen the conclusion of Theorem B by using Isaacs’ Bπ-theory.We also give an alternative proof of Theorem A by making use of this deeptheory. The results contained in Section 2.4 appear in a joint work of theauthor together with G. Navarro [NV15]. Finally, in Section 2.5 we proveTheorem D.

2.2. Two criteria for monomiality

Let N Ÿ G and χ P IrrpGq. Let θ P IrrpNq be a constituent of χN . WriteT “ Gθ for the inertia subgroup of θ. By Clifford’s correspondence there isa unique ψ P IrrpT |θq such that ψG “ χ. Of course, Qpχq Ď Qpψq, but ingeneral these two fields are not equal. The semi-inertia subgroup of θ isdefined as

T ˚ “ tg P G | θg “ θσ for some σ P GalpQpθqQqu.

Since T ď T ˚, then η “ ψT˚

P IrrpT ˚|θq also induces χ. Moreover Qpηq “Qpχq. Due to this fact, when dealing with character fields and normalsubgroups, the semi-inertia group is a useful tool.

Lemma 2.1. Let N Ÿ G and χ P IrrpGq. Let θ P IrrpNq be a constituentof χN . Write T and T ˚ for the inertia and the semi-inertia groups of θ. Ifψ P IrrpT |θq is the Clifford correspondent of χ, then QpψT˚q “ Qpχq.

Proof. See Lemma 2.2 of [NT10].

We do not need more preparation in order to prove Theorem A, whichwe restate below. We recall that if H ď G, then

coreGpHq “č

gPG

Hg.


22 2.2. Two criteria for monomiality

Theorem 2.2. Let G be a solvable group. Let χ P IrrpGq. If χp1q isodd and pχp1q, fχq “ 1, then there exists a subgroup U ď G and a linearcharacter λ of U such that λG “ χ. Moreover, if µ is a linear character ofsome subgroup W ď G such that µG “ χ, then there exists some g P G suchthat W “ Ug and µ “ λg.

Proof. First, we prove by induction on |G| that χ is monomial. Weprove it in a series of steps.

Step 1. We may assume χ is faithful and that there are no proper sub-groups H ă G and ψ P IrrpHq such that ψG “ χ and Qpψq “ Qpχq.

Let K “ kerpχq. If K ą 1, all the hypotheses hold in GK so byinduction we are done. Assume there exists H ă G and ψ P IrrpHq withQpψq “ Qpχq such that ψG “ χ. Then the degree of ψ divides the degreeof χ and fψ “ fχ. By induction hypothesis ψ is monomial, and thus χ ismonomial.

Step 2. FpGq “ś

p-χp1qOppGq.

Let p be a prime. Suppose that p divides χp1q. In particular p is odd,and p does not divide fχ. Let M be a normal p-subgroup of G. SinceQpχq Ď Qfχ we have that Q|M | X Qpχq “ Q. Hence χM is rational-valued.

If ξ P IrrpMq, then rχM , ξs “ rχM , ξs. Since χp1q is odd, there exists a realirreducible constituent ξ of χM . Since |M | is odd, we have that ξ “ 1M ,by Theorem 1.19. By Step 1, we know that χ is faithful and we concludeM “ 1.

Step 3. F “ FpGq is abelian.Let M be a normal p-subgroup of G, where p does not divide χp1q. It

then follows that the irreducible constituents of χM are linear. Let λ PIrrpMq be under χ. We have that M 1 ď kerpλgq “ kerpλqg for every g P G.Then M 1 ď coreGpkerpλqq ď kerpχq “ 1, so that M is abelian. Hence F isabelian by Step 2.

Step 4. Let N Ÿ G and let θ P IrrpNq be under χ. Let g P G. Thenθg “ θσ for some σ P GalpQpθqQq. Also θ is faithful.

Let T “ Gθ be the stabilizer of θ in G, and write T ˚ for the semi-inertiasubgroup of θ. Recall T ˚ “ tg P G | θg “ θσ for some σ P GalpQpθqQqu.By Lemma 2.1, if ψ P IrrpT |θq is the Clifford correspondent of χ, then

η “ ψT˚

P IrrpT ˚q induces χ and Qpηq “ Qpχq. By Step 1, we have thatT ˚ “ G, and so every G-conjugate of θ is actually a Galois conjugate. Thuskerpθgq “ kerpθq for every g P G. It follows that kerpθq Ÿ G and kerpθq iscontained in kerpχq by Clifford’s theorem. So θ is faithful by Step 1.

Final Step. If λ P IrrpF q is under χ, then λG “ χ.Let λ P IrrpF q be under χ. If y P G is such that λy “ λ, then we have

rx, ys P kerpλq for every x P F . By Step 4, λ is faithful, so the element ycentralizes F . Since F is self-centralizing, see 6.1.4 of [KS04], necessarilyy P F . We have proved Gλ “ F . Thus λG is irreducible and thus λG “ χ.This finishes the proof that χ is monomial.



Now, we work by induction on |G| to show that if U and V are subgroupsof G and λ P IrrpUq and µ P IrrpV q are linear such that λG “ χ “ µG, thenthere is some g P G such that V “ Ug and µ “ λg. Since K “ kerpχq ďcoreGpkerpλqq

Ş

coreGpkerpµqq we may assume that χ is faithful, for if K ą 1then we can work in GK. If p is a prime not dividing χp1q, then OppGqis contained in both U and V , because |G : U | “ χp1q “ |G : V |. LetF “ FpGq. By Step 2 (for which we only required that χ is faithful), wehave that

F “ź

p-χp1qOppGq ď U X V .

Now λF and µF lie under χ, so that µF “ pλF qg for some g P G by Clifford’s

theorem. We may assume that µF “ ν “ λF , by replacing the pair pU, λqby some G-conjugate. Thus U and V are contained in T “ Gν and also inT ˚, the semi-inertia subgroup of ν. Since λG and µG are irreducible, alsoλT and µT are irreducible. By uniqueness of the Clifford correspondent, wededuce that λT “ µT . In particular λT

˚

“ µT˚

“ ψ P IrrpT ˚|νq. We knowthat Qpψq “ Qpχq, again using Lemma 2.1. If T ˚ ă G, then the resultfollows by induction. Hence, we may assume T ˚ “ G. In particular, arguingas in the first part of the proof, we conclude that νG “ χ. This implies thatU “ F “ V and the theorem is proven.

Under the hypothesis of Theorem 2.2, it can also be proved that in factχ is supermonomial, that is, that every character inducing χ is monomial.The arguments are the same as in the proof of Theorem 2.2.

Let G be solvable. Let χ P IrrpGq be an odd degree character. Iffχ “ 2a for some a ě 0, then Theorem 2.2 guarantees that the characterχ is monomial. Let p be a prime. We can prove a p-version of the latterstatement for p-solvable groups, alas an oddness condition is still necessary.This is our Theorem B mentioned in the introduction, which appears in thissection as Theorem 2.6.

The following Lemma follows from an standard argument and will beoften used along this work.

Lemma 2.3. Let N Ÿ G and P P SylppGq. Let χ P IrrpGq have p1-degree.Then there is a P -invariant θ P IrrpNq under χ, and any two of them areNGpP q-conjugate. In particular, if NGpPNq “ PN , then θ is unique.

Proof. Let θ1 P IrrpNq be under χ, let T1 “ Gθ1 be the stabilizer ofθ1 in G and let ψ1 P IrrpT1|θ1q be the Clifford correspondent of χ over θ1.Since χ has p1-degree, we have that |G : T1| is not divisible by p, and then

P h´1ď T1 for some h P G. Then P ď T “ Gθ, where θ “ pθ1q

h P IrrpNq.Also, if η P IrrpNq is also P -invariant under χ, then by Clifford’s theoremwe have that ηg “ θ for some g P G. Then P, P g ď T , and thus P gt “ Pfor some t P T by Sylow Theory. Now ηgt “ θt “ θ, and hence η and θ areNGpP q-conjugate. The second part easily follows.


24 2.2. Two criteria for monomiality

Remark 2.4. Let π be a set of primes, let G be a π-separable group, letH be a Hall π-subgroup of G and let N Ÿ G. If χ P IrrpGq has π1-degree,then there is some H-invariant θ P IrrpNq under χ and any two of them areNGpHq-conjugate. The argument is analogous to the one given in the proofof Lemma 2.3

The following easy argument will be used sometimes in this chapter.

Lemma 2.5. Let P be a p-group. Suppose that P acts coprimely on agroup N and that CN pP q has odd order. Let θ P IrrpNq be P -invariant. Ifθ is real, then θ “ 1.

Proof. let θ P IrrpNq be By the Glauberman correspondence (see The-orem 1.17) with respect to the coprime action of the p-group P on N , wehave a natural bijection

˚ : IrrP pNq Ñ IrrpCN pP qq ,

where IrrP pNq is the set of P -invariant irreducible characters of N . Sinceθ is real, then also θ˚ is real. But CN pP q has odd order by hypothesis.Thus θ˚ “ 1 by Theorem 1.19 on real characters of groups of odd order, andtherefore θ “ 1N since ˚ is bijective and p1N q

˚ “ 1CN pP q.

Now, we can prove Theorem B.

Theorem 2.6. Let G be a p-solvable group for some prime p. Let P PSylppGq. Assume that NGpP qP has odd order. If χ P Irrp1pGq takes valuesin Qpa, then there exist a subgroup U and linear character λ of U withQpλq Ď Qpa such that χ “ λG. Also, if there exist J ď G and ψ P IrrpJqwith ψG “ χ, then ψ “ τJ for some linear character τ of a subgroup W ď J ,W “ Ug and τ “ λg for some g P G. In particular Qpψq Ď Qpa.

Proof. We first show the existence of U and λ. We argue by inductionon |G|. Of course, we may assume that G is non-abelian, so G1 ą 1.

Step 1. If N Ÿ G, θ P IrrpNq, g P G and σ P GalpQpθqQq, then we havethat pθσqg “ pθgqσ. In particular, the stabilizer of θ in G is the stabilizer ofθσ in G.

This immediately follows from the corresponding definitions.

Step 2. Suppose that N Ÿ G and let θ P IrrpNq be P -invariant under χ.If the complex conjugate θ of θ is also an irreducible constituent of χN , thenθ “ θ.

By Step 1, we have that θ is also P -invariant, and therefore there existsg P NGpP q such that θ “ θg, by Lemma 2.3. Now, g2 fixes θ (also usingStep 1). Now since NGpP qP has odd order by hypothesis, we concludethat xgP y “ xg2P y. Therefore g fixes θ and θ “ θ is real.

Step 3. We have that N “ Op1pGq ď kerpχq.By Lemma 2.3, let θ P IrrpNq be P -invariant under χ, let T be the

stabilizer of θ in G and let ψ P IrrpT q be the Clifford correspondent of χover θ. We prove now that θ is real. By Step 2, it suffices to show that θ is



under χ. Notice that QpχN q Ď Qpa X Q|N | “ Q. In particular, χN is real-

valued and so θ also lies under χ. Notice that CN pP q “ NN pP q ď NGpP qPhas odd order by hypothesis. By Lemma 2.5, we have that θ “ 1N . HenceOp1pGq ď kerpχq, as claimed.

Step 4. The character χ is faithful. In particular N “ 1.Let L “ kerpχq. Then NGLpPLLqpPLLq – NGpP qPNLpP q has

odd order. If L ą 1, then the existence of suitable U and λ is readilyobtained by applying the inductive hypothesis in GL.

Step 5. G is not a p-group.Otherwise, since p does not divide χp1q, we would have χ is linear. In

this case, there is nothing to prove.

Step 6. M “ OppGq ą 1 is abelian.Let ν P IrrpMq be under χ. Since χ has p1-degree, then we have that

ν is linear. Thus M 1 is contained in the kernel of every G-conjugate of ν.Since χ is faithful, we deduce that M is abelian. By Step 4, Op1pGq “ 1 soM ą 1.

Step 7. Let K be a minimal normal subgroup of G contained in G1. Letµ P IrrpKq be P -invariant under χ. Then Qpµq Ď Qpa.

By Lemma 2.3, there exists a P -invariant µ P IrrpKq lying under χ. ByStep 4 K is an elementary abelian p-group. Hence µ is linear and Qpµq Ď Qp.If a ě 1, then Qpµq Ď Qpa . Otherwise, χ is rational. Hence µ lies under χ.By Step 2 µ is real, hence rational.

Step 8. Let I be the stabilizer of µ in G. Then I ă G and µ extends toI.

Let QK be s Sylow q-subgroup of IK. If p ‰ q, then µ extends to Qbecause K is a p-group. If q “ p, then I is a p-group. Since χ has p1-degree,then χQ has some linear constituent, and hence µ extends to Q. It followsthat µ extends to I. Since µ is linear and χ faithful, then K Ď I 1, and thusI ă G.

Final Step. Let ψ be the Clifford correspondent of χ with respect to µ.Since both χ and µ have values in Qpa (use Step 7), then also ψ has valuesin Qpa . Since ψp1q divides χp1q, we have that ψp1q is a p1-number. Also Icontains P and the oddness condition still holds in I. By Step 8, we havethat I ă G, so by induction hypothesis, there exists U ď I and a linearcharacter λ of U with values in Qpa such that ψ “ λI . Also λG “ χ.

Now, suppose that J ď G and ϕ P IrrpJq with ψG “ χ. Then |G : J | is ap1-number and so K ď J . Thus ϕ lies over some G-conjugate of µ. We mayreplace the pair pJ, ϕq by some G-conjugate and assume that ϕ actually liesover µ. Let S “ I X J “ Jµ and let η P IrrpSq be the Clifford correspondentof ϕ with respect to µ. Then ϕ “ ηJ , so that ηG “ ϕG “ χ. This impliesthat ηI is irreducible, lies over µ and induces χ. By the uniqueness of theClifford correspondent ηI “ ψ. Since the character ψ “ λI and λ is a linearcharacter of U ď I with values in Qpa , by the inductive hypothesis applied


26 2.3. Certain character correspondences

in I, we get that η “ τS , where τ is a linear character of a subgroup W ď Isuch that the pairs pU, λq and pW, τq are conjugate in I.

Remark 2.7. Notice that we have actually proven that the character χas in Theorem 2.6 is supermonomial.

Remark 2.8. Let χ be as in Theorem 2.6. Then fχ “ pb, where b ě 0.Suppose that χ “ ψG for some ψ P IrrpHq and H ď G. By Theorem 2.6,Qpψq Ď Qpb , so that fψ divides pb. By the induction formula, we have thatQpχq Ď Qpψq. This implies that fχ divides fψ. Hence fψ “ fχ.

Remark 2.9. Let π be a set of primes, let G be a π-separable groupand let H be a Hall π-subgroup of G. We can mimic the proof Theorem 2.6with π and H playing the role of p and P to get a π-version of this result(use Remark 2.4 and Hall theory in π-separable groups).

2.3. Certain character correspondences

Let G be a finite group, let p be a prime and let a ě 0. How many p1-degreeirreducible characters does G have with field of values contained in Qpa? Itdoes not seem easy at all how to answer this question in general. However,if G is p-solvable and NGpP qP has odd order, then this number can becomputed locally. We write XpapGq “ tχ P Irrp1pGq | Qpχq Ď Qpau. Weare going to prove that there exists a canonical bijection from XpapGq ontoXpapNGpP qq. The fact that there exists such a canonical bijection followsby using the natural correspondences Irrp1pGq Ñ Irrp1pNGpP qq constructedby Isaacs (in the case where p “ 2) and Turull (in the case where |NGpP q|is odd) (see [Isa73] and [Tur08] for these highly non-trivial theorems). Byusing the fact that XpapGq consists of monomial characters, see Theorem 2.6,and the main result of [Isa90] it is easier to construct a canonical bijectionXpapGq Ñ XpapNGpP qq, as we are going to show.

Lemma 2.10. Let U ď G and let p be a prime. Let λ be a linear characterof U such that χ “ λG P IrrpGq. Let P be a Sylow p-subgroup of U and writeN “ NGpP q. Then

(a) pλUXN qN is irreducible.

(b) Suppose µG “ ϕ P IrrpGq for some linear character µ of a subgroupL ď G which contains P . If pλUXN q

N “ pµLXN qN then χ “ ϕ.

Proof. See the proof of Lemma 2.3 of [Isa90].

Let G be p-solvable and let P P SylppGq. Suppose that NGpP qP hasodd order. If χ P XpapGq, then by Theorem 2.6 there exist U ď G anda linear IrrpUq with Qpλq Ď Qpa such that λG “ χ. Write N “ NGpP qand ϕ “ pλNXU q

N . Then Lemma 2.10 guarantees ϕ P IrrpNq. Clearlyϕ P XpapNq. In order to see that the map XpapGq Ñ XpapNq given byχ ÞÑ ψ is a well-defined bijection we need some preliminary results.



Let G be p-solvable. We recall how to extend a linear character λ froma Sylow p-subgroup of G to a subgroup M in a maximal way guaranteeingthat an extension of λ to M induces an irreducible character of G.

Theorem 2.11. Let G be a p-solvable group and let P P SylppGq. Sup-pose that λ P IrrpP q is linear, then there exists a unique subgroup M ofG containing P and maximal such that λ extends to M . Moreover, ifµ P IrrpMq extends λ and opµq is a power of p, then µG P IrrpGq.

Proof. For the first part see Corollary 2.2 of [IN08]. The second partis Theorem 3.3 of [IN08]

We will also need the following Lemma.

Lemma 2.12. Let G be a group and let p be a prime. Let P P SylppGqand assume that P Ÿ G and GP has odd order . If λ P IrrpP q is linear andQpλq Ď Qpa, then there exists a unique χ P IrrpGq over λ with Qpχq Ď Qpa.

Proof. First note that every ψ P IrrpGq lying over λ has p1-degree.

By Lemma 1.13, let pλ be the canonical extension of λ to T “ Gλ. We

have that oppλq “ opλq. Hence pλG P XpapGq. Now, suppose ψ P XpapGqlies over λ. Then ψ “ τG for some τ P IrrpT |λq. Using Theorem 1.12

τ “ βpλ for some β P IrrpT P q. Since T P has odd order, by Theorem1.19 if β is real then β “ 1. Therefore, if β is real, then ψ “ χ. Assumeβ is not real and let σ P GalpQpβqQq be the complex conjugation. SinceQpβq Ď Q|T P | and |T P | is not divisible by p, we can extend σ to anelement of GalpQpapβqQpaq, by the natural irrationalities theorem of Galois

theory. We have, β “ βσ ‰ β. Since ψσ “ ψ and λσ “ λ, it must be

τσ “ βσpλσ “ βpλ “ βpλ “ τ . Using Theorem 1.12 again, βpλ “ βpλ impliesβ “ β, a contradiction.

We are ready to prove Theorem C, which we restate below.

Theorem 2.13. Let G be a p-solvable group and let P P SylppGq. WriteN “ NGpP q and assume that NP has odd order. Define a map

Ω: XpapGq Ñ XpapNq

in the following way: If χ P XpapGq, choose a pair pU, λq where P ď U ď Gand λ P IrrpUq linear with Qpλq Ď Qpa such that λG “ χ, then set Ωpχq “pλUXN q

N . Then Ω is a bijection.

Proof. Let χ P XpapGq, by Theorem 2.6 we know that there exists apair pU, λq where U ď G, λ is a linear character of U with Qpλq Ď Qpa

and λG “ χ. As χp1q “ |G : U | is a p1-number, U contains a Sylowp-subgroup of G. We may assume P ď U maybe replacing pU, λq by aconjugate pair. By Lemma 2.10(a), we have that ϕ “ pλUXN q

N P IrrpNq.Also pλUXN q

N p1q “ |N : U X N | is a p1-number, and Qpϕq Ď Qpλq Ď Qpa .Then, we get ϕ P XpapNq. In this situation, we will say that ϕ arises from


28 2.3. Certain character correspondences

χ. Using Lemma 2.10(b) we know that ϕ cannot arise from any ψ P XpapGqdifferent from χ. We have seen that in case Ω is well-defined, it is injective.

In order to prove that Ω defines a bijection, we need to show that Ω iswell defined and surjective. First, we show that ϕ is the only element thatarises from χ. Suppose that pW, νq is another pair such that P ď W ď G,ν is linear and νG “ χ. Then by Theorem 2.6, we have that W “ Ug andν “ λg for some g P G. Thus, P, P g ď W . By Sylow Theory, there existst P W such that P “ P gt. Also W “ Ugt and ν “ λgt. Hence we mayassume g P N . Then, it suffices to show that pλgUgXN q

N “ pλUXN qN . Write

µ “ λgUgXN . For every n P N we have that

µN pnq “1

|U XN |

ÿ

xPN

9µpxnx´1q

“1

|U XN |

ÿ

xPN

9λUXN pgxnpgxq´1q

“1

|U XN |

ÿ

yPN

9λUXN pyny´1q

“ pλUXN qN pnq.

Finally, we prove that Ω is surjective. Let θ P XpapNq, let λ P IrrpP q beunder θ. Then, λ is linear and opλq “ |P : kerpλq|. By Theorem 2.11 thereexists M ď G containing P maximal such that λ extends to M . Further-more, M “ PU where P normalizes U and P X U “ kerpλq. Then, we canchoose the unique extension ν of λ with opνq “ opλq. Again using Theorem2.11, we have that νG P Irrp1pGq. We need to show that Qpνq Ď Qpa . Itsuffices to see that Qpλq Ď Qpa . We distinguish the cases where a “ 0 anda ą 0.

Case a “ 0. In this case θ is rational. Then both λ and λ are under θ. ByClifford’s theorem λ “ λg for some g P N , and we see that g normalizes T the

stabilizer of λ inN for the two actions commute. Then, λ “ λg “ pλqg “ λg2,

g2 P T . But NN pT qT has odd order and thus g P T . Hence λ takes realvalues, since λ is linear we conclude λ is rational.

Case a ą 0. We have that Qpλq Ď Qopλq where opλq “ pb. Suppose

pb ą pa and let 1 ‰ σ P GalpQpbQpaq. Since

|GalpQpbQpaq| “|GalpQpbQq||GalpQpaQq|

“pb´1pp´ 1q

pa´1pp´ 1q“ pbá,

σ has p-power order. Since χσ “ χ, by Clifford’s Theorem, λσ “ λg forsome g P G that normalizes T , the stabilizer of λ in N . We know that theaction by the automorphisms of G and Galois action on IrrpNq commute,

then gopσq P T . Since |NN pT qT | is a p1-number, it must be that g P T , andconsequently λσ “ λ. Now, we know that pνMXN q

N P XpapNq. If we showthat pνMXN q

N “ θ we will be done. This is true since by Lemma 2.12 thereis a unique element of XpapNq over λ.



The bijection Ω given by Theorem 2.13 is completely canonical. Inparticular, let α be an automorphism of G that fixes P . Let χ P XpapGq andλ P IrrpUq such that Ωpχq “ pλUXN q

N . Then we have that Ωpχαq “ Ωpχqα

because pλαUαXN qN “ ppλUXN q

N qα.

2.4. Certain monomial characters and their subnormalconstituents

As promised in the introduction, we use Isaacs Bπ-theory of π-separablegroups (see Section 1.4) to strengthen the conclusion of Theorem B. Themain feature of the use of this theory is that in the situation of Theorem Bwe can guarantee that such χ is not only monomial but also any subnormalconstituent of χ is monomial, see Theorem 2.15 below. The results containedin this section have been published in a joint work of the author with G.Navarro in [NV15].

We refer the reader to Section 1.4 for the definition and first propertiesof the Isaacs Bπ-characters.

Lemma 2.14. Suppose that G is a p-solvable group. Let P P SylppGq,and assume that NGpP qP has odd order. If α P IrrpGq is p1-special andreal, then α is the trivial character.

Proof. We argue by induction on |G|. Let N “ OppGq. By Proposition1.20, we have that N ď kerpαq. If N ą 1, then we apply induction in GN .Otherwise, let K “ Op1pGq. Then K ą 1. By Lemma 2.3, there is someP -invariant θ P IrrpKq under α, and any two of them are NGpP q-conjugate(since α is p1-special it has in particular p1-degree). Since α is real, then θis also under α, and therefore there is g P NGpP q such that θ “ θg. Nowg2 fixes θ, and since NGpP qP has odd order, we see that θ “ θ. Also, thefact that NGpP qP has odd order implies that that CKpP q has odd order.By Lemma 2.5, we have that θ “ 1K . Thus K ď kerpαq, and we applyinduction in GK.

We are ready to prove the main result of this section.

Theorem 2.15. Let p be a prime, let G be a p-solvable group, and letP P SylppGq. Let χ P IrrpGq be such that p does not divide χp1q and suchthat Qpχq Ď Qpa for some a ě 0. If |NGpP qP | is odd, then χ P BppGq. Inparticular, if N Ÿ ŸG and θ is an irreducible constituent of χN , then θ ismonomial.

Proof. By Theorem 1.27, there exists a subgroup P ď W ď G anda p-special linear character λ P IrrpW q, such that: ψ “ λG P IrrpGq isa Bp-character, and pW,λq is a nucleus of ψ. Also, there is a p1-specialcharacter α P IrrpW q such that χ “ pλαqG. By Theorem 1.26, the pairpW,λαq is unique up to G-conjugacy. Now, let σ P GalpQ|G|Q|G|pq be the

unique Galois automorphism that complex conjugates the p1-roots of unityand fixes p-power roots of unity. Since χ and λ are fixed by σ, then we deduce


302.4. Certain monomial characters and their subnormal

constituents

that there is g P G such that pW g, λgαgq “ pW,λασq. Hence αg “ ασ byProposition 1.23. Since P, P g ďW , then P gw “ P for some w PW , and we

may assume that g P NGpP q. Also, αg2“ α, and therefore since NGpP qP

has odd order, we see that ασ “ α. Now, let H be a p-complement of W .Then

αH “ αH “ pασqH “ αH

and we deduce that α “ α, by using Proposition 1.22. Since NW pP qP hasodd order, by Lemma (2.1), we have that α “ 1W . Thus χ “ ψ P BppGqand χ is monomial. Now, to prove the second part of the theorem, usethat subnormal constituents of Bp-characters are Bp-characters by Theorem1.24, and the second part of Theorem 2.2 of [CN08] , that asserts that Bp-characters of p1-degree are monomial.

We obtain the following consequence, in which a global invariant of afinite group is calculated locally.

Corollary 2.16. Let p be a prime, let G be a p-solvable group, andlet P P SylppGq. Assume that NGpP qP has odd order. Then the numberof irreducible characters χ of G such that χp1q is not divisible by p andQpχq Ď Q|G|p is the number of orbits of the natural action of NGpP q on

P P 1.

Proof. Recall the notation from the previous section. If a is a non-negative integer we write XpapGq “ tχ P Irrp1pGq | Qpχq Ď Qpau. Let|P | “ pa. By Theorem 2.15, we have that XpapGq “ BppGq X Irrp1pGq andXpapNq “ BppNqX Irrp1pNq. By Theorem 2.2 and Corollary 2.3 of [CN08],we have that |XpapGq| “ |XpapNq|. We will show that |XpapNq| is equalto the number of orbits under the natural action of NGpP q on IrrpP P 1q.Let Λ be a complete set of representatives of the NGpP q orbits on P P 1. Ifθ P XpapGq, then by Clifford’s theorem θ lies over a unique λ P Λ. Hence|Xpa | ď |Λ|. Now, suppose that θ, θ1 P XpapNq lie over the same λ P Λ.Then there exist ψ,ψ1 P IrrpNλ|λq such that θ “ ψN and θ1 “ pψ1q

G. No-tice that ψ and ψ1 have p1-degree by the induction formula. By Theorem1.13 λ extends canonically to Nλ and by Theorem 1.12, we have that ψ “ βλand θ1 “ β1λ, where β, β1 P Irrp1pNλP q and λ is the canonical extension ofλ to Nλ. Since Qpθq, Qpθ1q and Qpλq are contained in Qpa , then also Qpψqand Qpψ1q are contained in Qpa . Hence β and β1 are rational valued. Sinceby assumption NP is odd, by Burnside’s theorem 1.19 we have that β “ β1

and therefore θ “ θ1.

As we have already mentioned, we can shorten the proof of our Theorem2.2 if we are willing to use Isaacs Bπ-theory.

Theorem 2.17. Suppose that G is solvable. Suppose that χ P IrrpGq hasodd degree. Suppose that pχp1q, fχq “ 1. Then χ is monomial.

Proof. Let π be the set of primes (possibly empty) dividing fχ. Thenχ has π1-degree. By Theorem 1.27, there exist a pair pW,γq, where W ď G



and γ P IrrpW q is π-special, and a π1-special character α P IrrpW q suchχ “ pαγqG. Also, γG “ ψ P BπpGq and pW,γq is a nucleus for ψ. Also byTheorem 1.26, the pair pW,γαq is unique up to G-conjugacy. Notice thatχp1q “ |G : W |αp1qγp1q implies that γp1q “ 1 and W contains a full Sylow2-subgroup of G.

We have that

Qpχq Ď Qfχ Ď Q|G|π .Since γ is π-special, Qpγq Ď Q|G|π . Now, let σ P GalpQ|G|π1 Qq. We may

extend σ to some σ P GalpQ|G|Q|G|πq. Then σ fixes χ and γ. ThuspW,γασq “ pW,γαqg for some g P G. In particular, g P NGpW q. By

Proposition 1.23, we have that γ “ γg and and ασ “ αg. Let β “ αNGpW q.Now, although β is not necessarily irreducible we have that Qpβq Ď Q|G|π1 .Moreover, since ασ “ αg, we have that β is fixed by σ and therefore β isrational valued (of odd) degree.

Now αNGpW q “ β “ β “ pαqNGpW q. It follows that some irreducibleconstituent ψ of β lies over α and α. Hence α “ αg for some g P NGpW q.Since NGpW qα “ W , we have that g2 P W , but |NGpW qW | odd impliesthat g PW . Then α is real, and by Gow’s theorem it is monomial. Hence χis monomial.

2.5. Feit’s Conjecture and p1-degree characters

Our aim in this section is to prove Theorem D of the introduction. We needa little preparation in order to do that. We will use Gajendragadkar specialcharacters (we refer the reader to Section 1.4). The following result will helpus to control fields of values under certain circumstances.

Lemma 2.18. Let G be a finite group, let q be a prime and let ζ be aprimitive q-th root of unity. Suppose that G is q-solvable and χ P IrrpGq isq-special. If χ ‰ 1, then ζ P Qpχq.

Proof. Let Q be a Sylow q-subgroup of G. By Lemma 1.22, we havethat ψ ÞÑ ψQ is an injection from the set of q-special characters of G intothe set IrrpQq. In particular Qpχq “ QpχQq. Of course χQ ‰ 1. Thus, wemay assume that G is a q-group. We also may assume that χ is faithfulby modding out by kerpχq. Choose x P ZpGq of order q. We have thatχxxy “ χp1qλ, where λ P Irrpxxyq is faithful. Hence λpxiq “ ζ for someinteger i. In particular ζ P Qpχq.

The following elementary observation is stated as a Lemma for thereader’s convenience.

Lemma 2.19. Suppose that λ is a linear character of a finite group, andlet P P SylppGq. Let NGpP q Ď H ď G and let ν “ λH . Then opλq “ opνq.

Proof. If λ “ 1G, then there is nothing to prove. We may assume λis non-principal and hence G1 ă G. We have that P Ď PG1 Ÿ G. By the


32 2.5. Feit’s Conjecture and p1-degree characters

Frattini argument, we have that G “ G1NGpP q “ G1H. Since G1 Ď kerpλqand kerpνq “ kerpλq XH, the result follows.

The proof of Theorem D requires the use of a magical character; thecanonical character associated to a character five defined by Isaacs in [Isa73].We summarize the properties of ψ below.

Let L Ď K Ÿ G with L Ÿ G and KL abelian. Let θ P IrrpKq andϕ P IrrpθLq. Suppose that θ is the unique irreducible constituent of ϕK (inthis case we say that ϕ is fully ramified with respect to KL or equivalentlythat θ is fully ramified with respect to KL) and ϕ is G-invariant. Thenwe say that pG,K,L, θ, ϕq is a character five.

Theorem 2.20. Let pG,K,L, θ, ϕq be a character five. Suppose thatKL is a q-group for some odd prime q. Then there exist a character ψ ofG with K Ď kerpψq and a subgroup U ď G such that

(a) UK “ G and U XK “ L;(b) ψpgq ‰ 0 for every g P G, ψp1q2 “ |K : L| and the determinantal

order of ψ is a power of q;(c) if K Ď W ď G, then the equation ξW “ ψW ξ0 for ξ P IrrpW |θq

and ξ0 P IrrpW XU |ϕq defines a one-to-one correspondence betweenthese two sets; and

(d) if K ĎW ď G, then ξ P IrrpW |θq and ξ0 P IrrpWXU |ϕq correspondin the sense of (c) if and only if ξG0 “ ψW ξ, where ψ denotes thecomplex conjugate of ψ.

(e) If KL is elementary abelian, then Qpψq Ď Qq.

Proof. For parts (a), (b) and (c) see Theorem 3.1 of [Nav02]. Part(d) follows from Corollary 9.2 of [Isa73] (since the complement U providedby [Nav02] is ”good” not only for GL but also for every W L whereK ĎW ď G). For part (e), by Theorem 9.1 of [Isa73] and the discussion atthe end of the page 619 of [Isa73], the values of the character ψ are Q-linearcombinations of products of values of the bilinear multiplicative symplecticform ! , "ϕ : K ˆ K Ñ Cˆ associated to ϕ (defined at the beginning ofSection 2 of [Isa73]). The values of ! , "ϕ are values of linear characters ofcyclic subgroups of KL. Since KL is q-elementary abelian, we do obtainthat Qpψq Ď F .

We can prove Theorem D, which we restate here.

Theorem 2.21. Let p be a prime and let G be a finite solvable group.Let χ P IrrpGq of degree not divisible by p, and let P P SylppGq. If χp1q isodd, then there exists g P NGpP qP

1 such that opgq “ fχ. In particular, theFeit number fχ divides |NGpP q : P 1|.

Proof. By the Amit-Chillag theorem [AC86], we may assume that pdivides |G|. We proceed by induction on |G|.

Let N Ÿ G. If θ P IrrpNq is P -invariant and lies under χ, then we mayassume that θ is G-invariant. Let ψ P IrrpGθ|θq be the Clifford correspondent



of χ. By the character formula for induction, Qpχq Ď Qpψq and χp1q “ |G :Gθ|ψp1q. Thus the character ψ satisfies the hypotheses of the theorem inGθ and fχ divides fψ. If Gθ ă G, then by induction there exists someg P NGθpP qP

1 ď NGpP qP1 (notice that the P -invariance of θ implies

P Ď Gθ) such that opgq “ fψ. Hence, some power of g has order fχ and wemay assume Gθ “ G.

We claim that we may assume that χ is primitive. Otherwise, supposethat χ is induced from ψ P IrrpHq for some H ă G. In particular, p doesnot divide |G : H| and so H contains some Sylow p-subgroup of G, whichwe may assume is P . Again by the character formula for induction, thedegree ψp1q is an odd p1-number and fχ divides fψ. By induction there isg P NHpP qP

1 ď NGpP qP1 such that opgq “ fψ. Thus some power of g has

order fχ, as claimed.

By Theorem 2.6 of [Isa81] the primitive character χ factorizes as aproduct

χ “ź

q

χq,

where the χq are q-special characters of G for distinct primes q. Let σ PGalpQ|G|Qpχqq. Then

ź

q

χσq “ź

q

χq.

By using the uniqueness of the product of special characters, see Proposition1.23, we conclude that χσq “ χq for every q. Hence fχq divides fχ forevery q, and since the fχq ’s are coprime also

ś

q fχq divides fχ. Notice that

Qpχq Ď Qpχq | qq Ď Qś

q fχqby elementary Galois theory. This implies the

equality fχ “ś

q fχq .

Now, consider K “ Op1,ppGq ă G. Notice that PK “ OppGq Ÿ G. Bythe Frattini argument G “ PKNGpP q “ KNGpP q. If K “ 1, then P Ÿ Gand we are done in this case. We may assume that K ą 1. Let KL bea chief factor of G. Then KL is an abelian p1-group. If H “ NGpP qL,then G “ KH and K XH “ L, by a standard group theoretical argument.Furthermore, all the complements of K in G are G-conjugate to H. Finally,notice that CKLpP q “ 1 using that H XK “ L.

We claim that for every q, there exists some q-special χ˚q P IrrpHq such

that fχ˚q “ fχq and χ˚q p1q is an odd p1-number.

If q P t2, pu, then λ “ χq is linear (because χ has odd p1-degree). Letλ˚ “ λH . Then λ˚ is q-special (since λ is linear and q-special, this isstraightforward from the definition) and fλ˚ “ fλ by Lemma 2.19.

Let q ‰ p be an odd prime and write η “ χq. We work to find someη˚ P IrrpHq of odd p1-degree with fη˚ “ fη. By Lemma 2.3, let θ P IrrpKq besome P -invariant constituent of ηK and let ϕ P IrrpLq be some P -invariantconstituent of ηL. By the second paragraph of the proof, we know that both


34 2.5. Feit’s Conjecture and p1-degree characters

θ and ϕ are G-invariant and hence ϕ lies under θ. By Theorem 6.18 of[Isa76] one of the following holds:

(a) θL “řti“1 ϕi, where the ϕi P IrrpLq are distinct and t “ |K : L|,

(b) θL P IrrpLq, or(c) θL “ eϕ, where ϕ P IrrpLq and e2 “ |K : L|.

Notice that the situation described in (a) cannot occur here, because ϕis G-invariant.

In the case described in (b), we have ϕ “ θL P IrrpLq. Then restrictiondefines a bijection between the set of irreducible characters of G lying overθ and the set of irreducible characters of H lying over ϕ (by Corollary(4.2) of [Isa86]). Write ξ “ ηH . By Theorem A of [Isa86], we knowthat ξ is q-special. We claim that Qpηq “ Qpξq. Clearly, Qpξq Ď Qpηq.If σ P GalpQpηqQpξqq, then notice that ϕ is σ-invariant because ξL is amultiple of ϕ. Now, ϕ is P -invariant, and because CKLpP q “ 1, there isa unique P -invariant character over ϕ (by Problem 13.10 of [Isa76]). Byuniqueness, we deduce that θσ “ θ. Now, ησ lies over θ and restricts toξ, so we deduce that ησ “ η, by the uniqueness in the restriction. ThusQpηq “ Qpξq. We write η˚ “ ξ.

Finally, we consider the situation described in (c). Since θL is not ir-reducible, then |K : L| is not a q1-group, by Theorem 1.10. Hence KL isq-elementary abelian and e is a power of q. By Theorem 2.20 (and usingthat all the complements of KL in GL are conjugate), there exists a (notnecessarily irreducible) character ψ of G such that:

(i) ψ contains K in its kernel, ψpgq ‰ 0 for every g P G, ψp1q “ e andthe determinantal order of ψ is a power of q.

(ii) if K Ď W ď G and ξ P IrrpW |θq, then ξWXH “ ψWXHξ0 for aunique irreducible character ξ0 of W XH.

(iii) The values of ψ lie on Qq.

In particular, ηH “ ψη0, so that η0 P IrrpH|ϕq (where we are viewing ψas a character of H). We claim that η0 is q-special. First notice thatη0p1q “ ηp1qe is a power of q. Now, we want to show that whenever Sis a subnormal subgroup of H, the irreducible consituents of pη0qS havedeterminantal order a power of q. Since pη0qL is a multiple of ϕ, which isq-special, we only need to control the irreducible constituents of pη0qS whenL Ď S Ÿ ŸH, by using Proposition 1.20. We have that K Ď SK Ÿ ŸG.Write

ηSK “ a1γ1 ` ¨ ¨ ¨ ` arγr,

where the γi P IrrpSKq are q-special because η is q-special and ai P N0. Byusing the property (ii) of ψ, we have that ηS “ ψSpη0qS also decomposes as

ηS “ a1ψSpγ1q0 ` ¨ ¨ ¨ ` arψSpγrq0

“ ψSpa1pγ1q0 ` ¨ ¨ ¨ ` arpγrq0q.



Since ψ never vanishes on G, we conclude that pη0qS “ a1pγ1q0`¨ ¨ ¨àrpγrq0.It suffices to see that oppγiq0q is a power of q for every γi constituent of ηSK .Just notice that

detppγiqSq “ detpψSpγiq0q

“ detpψSqpγiq0p1qdetppγiq0q

e.

Since opψq, opγiq, γip1q and e are powers of q, we easily conclude that alsothe determinantal order of pγiq0 is a power of q. This proves that η0 is q-special. We claim that Qpηq “ Qpη0q so that the two Feit numbers are thesame. Let ζ be a primitive q-th root of unity and write F “ Qpζq. Then thevalues of ψ lie in F . We next see that η and η0 are non-principal. This isobvious because θ and ϕ are fully ramified. Suppose that σ P GalpQ|G|F qstabilizes η. Then

ψη0 “ ψσησ0 “ ψησ0 .

Using that ψ is never zero, we conclude that ησ0 “ η0. Now, by part (d) ofTheorem 2.20, we have that ξ and ξ0 correspond (as in part (c) of Theorem2.20) if and only if pξ0q

G “ ψξ. Hence, if σ P GalpQ|G|F q and ησ0 “ η0, then

ψη “ pη0qG “ pησ0 q

G “ ψησ (because also Qpψq Ď F ). This implies againthat ησ “ η. By Galois theory, we have that F pηq “ F pη0q. By Lemma2.18, this implies Qpηq “ Qpη0q. We set η˚ “ η0. The claim follows.

Now, we define χ˚ “ś

q χq˚ which has odd p1-degree. The character

χ˚ is irreducible by Proposition 1.23. Also fχ˚ “ś

q χ˚q as in the fourth

paragraph of this proof. Hence

fχ˚ “ź

q

fχq˚ “ź

q

fχq “ fχ.

By the inductive hypothesis, there exists g P NHpP qP1 ď NGpP qP

1 suchthat opgq “ fχ˚ and we are done.


CHAPTER 3

McKay natural correspondences of characters

3.1. Introduction

Let p be a prime. Recall that for a finite group G, we denote by Irrp1pGqthe set of the irreducible characters χ P IrrpGq of G which have degreeχp1q not divisible by p. The McKay conjecture, one of the main problemsin the Representation Theory of Finite Groups, asserts that |Irrp1pGq| “|Irrp1pNGpP qq|, where P P SylppGq. It is suspected that in general no choice-free correspondence can exist between Irrp1pGq and Irrp1pNGpP qq, at leastthere does not exist one commuting with Galois action. For instance, ifG “ GL2p3q, p “ 3 and P P SylppGq, then all the irreducible characters ofNGpP q are rational-valued, while G has characters of degree 2 which are notrational-valued.

A key case to consider is when NGpP q “ P . If NGpP q “ P , thenMcKay conjecture predicts the existence of a bijection between Irrp1pGq andIrrpP P 1q. We prove that much more is happening for p odd.

Theorem E. Let G be a group, let p be an odd prime and let P P

SylppGq. Suppose that P “ NGpP q. If χ P Irrp1pGq, then

χP “ χ˚ `∆ ,

where χ˚ P IrrpP q is linear and ∆ is either zero or ∆ is a character whoseirreducible constituents have all degree divisible by p. Furthemore, the mapχ ÞÑ χ˚ is a natural bijection Irrp1pGq Ñ IrrpP P 1q.

Theorem E was proved for p-solvable groups in [Nav07] for any prime p,although it was suspected long time ago that it should hold in general for oddprimes. For p “ 2, Theorem E is not true, the symmetric group S5 providesa counterexample. However, E. Giannelli [Gia16] found canonical bijectionsfor p “ 2 and Sn. (This bijection cannot be described by restriction unlessn “ 2k.) We also mention that in [GKNT16] the authors provide a differentcanonical bijection for p “ 2 and Sn as well as for GLnpqq and GUnpqq (forq odd).

We will also work in greater generality. Instead of assuming that P P

SylppGq is self-normalizing, we assume that NGpP q “ CGpP qP (namelyNGpP q is p-decomposable). We prove that a bijection as in Theorem E doesexist but at the moment only between characters in the principal blocks(again for p odd). Recall that a character χ P IrrpGq lies in the principal(p-)block B0pGq of G if

ř

xPG0 χpxq ‰ 0, where G0 is the set of elements of

37


G of order not divisible by p. If we further assume that G is p-solvable, thenwe can prove the following (with no restriction on p).

Theorem F. Let G be a finite p-solvable group, and let P P SylppGq.Suppose that NGpP q “ PCGpP q, and let N “ Op1pGq. Let IrrP pNq be theset of P -invariant characters θ P IrrpNq. Then, for every θ P IrrP pNq andλ P IrrpP P 1q linear, there is a canonically defined character

λ ‹ θ P Irrp1pGq .

Furthermore, the map

IrrpP P 1q ˆ IrrP pNq Ñ Irrp1pGq

given by pλ, θq ÞÑ λ ‹ θ is a bijection. As a consequence, if θ˚ P IrrpCN pP qqis the Glauberman correspondent of θ P IrrP pNq (see Theorem 1.17), thenthe map

λˆ θ˚ ÞÑ λ ‹ θ

is a natural bijection Irrp1pNGpP qq Ñ Irrp1pGq. Also, if θ “ 1N and λ PIrrpP P 1q, then λˆ θ˚ is the unique linear constituent of pλ ‹ θqNGpP q.

This chapter is structured in the following way: We begin by studyingsome character correspondences in the groups PSL2p3

3aq for a ě 1 in Section3.2. There is a good reason for that: these groups appear as the onlynon-abelian composition factors of groups with a self-normalzing Sylow p-subgroup for odd p (see [GMN04]). In Section 3.3 we prove the followingextension theorem which is key to prove Theorem E.

Theorem. Let N Ÿ G. Let p be an odd prime and let P P SylppGq.Assume that NGpP q “ P . If χ P Irrp1pGq and θ P IrrpNq lies under χ, thenθ extends to Gθ.

In Section 3.4 we prove Theorem E and we present an application to char-acterize groups with self-normalizing Sylow p-subgroups for odd p. In Sec-tion 3.5, we consider the case where the normalizer of the Sylow p-subgroupis p-decomposable.

We will start by proving a key group theoretical result, which extends aclassical work of J. Thompson (see Theorem 3.14 of [Isa08]).

Theorem. Let G be a group, let p be a prime, and let P P SylppGq.Suppose that NGpP q “ P ˆ X. If p is odd or G is p-solvable, then X ď

Op1pGq. In particular, if NGpP q “ PCGpP q, then Op1pNGpP qq ď Op1pGq.

After that we prove there exist character correspondences as in TheoremE between characters in the principal blocks. In the last section, we proveTheorem F.

All the results contained in this chapter, unless otherwise stated, appearin a joint work of the author together with G. Navarro and P. H. Tiep[NTV14].


3. McKay natural correspondences of characters 39

3.2. Character correspondences associated with PSL2p33aq

By the main result of [GMN04], if a group G has a self-normalizing Sylowp-subgroup for some odd prime p, then either G is solvable or p “ 3 and Ghas a composition factor of type PSL2p3

3aq where a ě 1. In the case whereG is not solvable, it is easy to show that all non-abelian composition factorsof G are of type PSL2p3

3aq with a ě 1.

Lemma 3.1. Let G be a group. Let p be an odd prime. Suppose thatNGpP q “ P for some P P SylppGq. If KL is a non-abelian composition

factor of G, then KL is of type PSL2p33aq with a ě 1.

Proof. By Corollary 1.2 of [GMN04], the claim holds when G is sim-ple. We proceed by induction on |G|. Let N be a minimal normal subgroupof G. Then 1 ă |N | ă |G|. Since both K XN and L are normal subgroupsof K, we have that L ď pK X NqL Ÿ K. However, KL simple yieldspK XNqL “ L or pK XNqL “ K.

Suppose pK X NqL “ L. Then KL – KNKL is a nonabelian com-position factor of GN . Since NGN pPNNq “ NGpP qNN “ PNN , by

induction hypothesis KL is of type PSL2p33aq with a ě 1.

Suppose that pK X NqL “ K. Then KL – pK X NqpL X Nq is anon-abelian composition factor of G0 “ PN . Therefore N is a non-abelianminimal normal subgroup. We can write N “ T1 ˆ ¨ ¨ ¨ ˆ Tr, where the Ti’sare non-abelian simple groups transitively permuted by G and consequentlyall isomorphic. Also, the Ti’s are the composition factors of N up to isomor-phism by the Jordan-Holder Theorem. Thus KL – Ti for some i. SinceNG0pP q “ P , if G0 ă G, then we are done by induction. We may assumethat G “ PN . By the main result of [GMN04], some composition factorof G, thus some composition factor Tj of N is of type PSL2p3

3aq with a ě 1.Hence KL is also of this type and we are done.

Due to these facts, the groups of type PSL2p33aq with a ě 1 play an

important role in the proof of our Theorem E. The description of the be-havior of the character theory of groups of type PSL2pqq under the actionof automorphisms was accomplished in Section (15B) of [IMN07]. In orderto make this chapter as self-contained as possible, this section is devoted todescribe how does the character theory of PSL2p3

3aq behave under the ac-tion of their field automorphisms using the background of Section 1.5. (Wewill use results contained in Section 1.5 without specific reference).

First we need the following result about some special actions of p-groupson groups of type PSL2pp

paq.

Lemma 3.2. Let S “ PSL2pqq, where q “ ppa, for some odd prime p

and a ě 1. Embed S Ÿ AutpSq “ A. Let P ď A be a p-subgroup such thatP X S “ Q P SylppSq. Suppose that P acts on S with CNSpQqQpP q “ 1.Then p “ 3 and P P SylppAq.


40 3.2. Character correspondences associated with PSL2p33aq

Proof. We proceed in a series of steps.Step 1. We may assume that Q is the Sylow p-subgroup Q1 of S induced

from upper unitriangular matrices.

We have that Q “ Qs1 for some s P S. Then Q1 “ P s´1XSŸ P s

´1“ P1

and P1 acts on S with CNSpQ1qQ1pP1q “ pCNSpQqQpP qq

s´1“ 1.

Step 2. We describe the Sylow p-subgroups of A and NApQq.Write F “ Fq. Let ϕ be the Frobenius automorphism of F . Then, ϕ

induces an automorphism of S and xϕy ď A has order f “ pa, see Section1.5.1. By Step 1, the group xϕy normalizes Q, and so Qxϕy is a subgroupof A of order pa ¨ q (we are using that xϕy X S “ 1). Hence Qxϕy P SylppAq,

because |A| “ paqpq2 ´ 1q. Write H “ NApQq. We claim:

(a) SylppAq “ tpQxϕyqs | s P Su,

(b) SylppHq “ tpQxϕyqt | t P NSpQqu.

We first prove (a). Since AS is abelian, we have that Sxϕy Ÿ A and Qxϕyis a Sylow p-subgroup of Sxϕy. By Frattini’s argument (see 3.2.7 of [KS04]for instance)

A “ SxϕyNApQxϕyq “ SNApQxϕyq,

hence the description in (a) follows. To prove (b), we proceed analogously.Since Q P SylppSq and SŸ A, we have that A “ SNApQq “ SH by Frattini’sargument. In particular, HNSpQq is abelian. We have that xϕy normalizesNSpQq. Hence NSpQqxϕyŸH and using Frattini’s argument one more time,we conclude

H “ NSpQqxϕyNHpQxϕyq “ NSpQqNHpQxϕyq.

Step 3. We may assume P ď Qxϕy. In particular P “ Qxϕey for someinteger e.

Since P ď NApQq is a p-subgroup, then P ď pQxϕyqt for some t PNSpQq, by Step 2. Let s “ t´1. We have Qs “ Q ď P s ď Qxϕy. We mayreplace P by P s because P s acts on S stabilizing Q and

CNSpQqQpPsq “ pCNSpQqQpP qq

s “ 1.

Thus, we may assume P ď Qxϕy. Hence P “ P X Qxϕy “ QpP X xϕyq “Qxϕey, for some integer e.

Final step. We conclude P “ Qxϕy is a full Sylow p-subgroup of A.By Step 3, we know that P “ Qxϕey for some integer e. Consider ϕe

both as a Galois automorphism of F and as an automorphism of S. Supposethat ϕe fixes an element a P Fˆ. Let dpaq “ diagpa, a´1q P NSpQq. By easy

matrix computations, for every x “

ˆ

1 b0 1

˙

P Q and for every integer k,

we have that

dpaq´1dpaqxϕek“

ˆ

1 bpekpa´ a´1q

0 1

˙

P Q.



This proves that dpaq P CNSpQqQpP q. By hypothesis CNSpQqQpP q “ 1, sothat dpaq P Q. This means that dpaq is the identity of S. Thus a “ ˘1 P F .Hence, the subfield of F fixed by ϕe is exactly F3. Since ϕe P GalpF Fpq,we conclude that p “ 3 and xϕey “ xϕy, by Galois Theory.

Let S “ PSL2pqq, where q “ 33a for some a ě 1. We shall also needthe following result about the automorphisms of S that centralize a Sylow3-subgroup of S.

Lemma 3.3. Let S “ PSL2pqq, where q “ 33a for some a ě 1.WriteA “ AutpSq. Let P P Syl3pAq. Write Q “ P X S P Syl3pSq. Suppose thatY ď A is a 31-subgroup that centralizes Q. Then Y “ 1.

Proof. We may assume that Q is the Sylow 3-subgroup of S inducedfrom upper unitriangular matrices. Since A “ PGL2pqq ¸ xϕy, where xϕy isthe group of field automorphisms of S with opϕq “ 3a, and Y is a 31-subgroupof A, we have that Y ď PGL2pqq. The result follows since CPGL2pqqpQq “Q.

Lemma 3.4. Let G be a group and let S ď G. Suppose that S “ PSL2pqq,where q “ 33a and a ě 1. Write p “ 3 and let P be a p-subgroup of Gsuch that Q “ P X S P SylppSq. Suppose further that P acts on S withCNSpQqQpP q “ 1. If α P Irrp1pSq is P -invariant, then

αQ “ α˚ `∆,

where α˚ is P -invariant and no irreducible constituent of the character ∆ isP -invariant. Moreover, the map α ÞÑ α˚ defines a bijection between the setof irreducible character of S of p1-degree fixed by P and the set of irreduciblecharacters of Q fixed by P .

Proof. We proceed in a series of steps. Let F “ Fq. We write ϕ todenote the Frobenius automorphism of the field F .

Step 1. We may assume Q is the Sylow p-subgroup Q1 of S induced fromupper unitriangular matrices.

We have that Qs “ Q1 for some s P S. Then Q1 “ P sXSŸ P s “ P1 andP1 acts on S with CNSpQ1qQ1

pP1q “ pCNSpQqQpP qqs “ 1. Also IrrP1pQ1q “

pIrrP pQqqs.

Step 2. We may work in AutpSq and we may assume that P “ Qxϕy,where we identify ϕ with the field automorphism of S corresponding to ϕ asin Section 1.5.

Write A “ AutpSq. Let γ : P Ñ A be the homomorphism defined byconjugation by P . Write P1 “ γpP q ď A. Embed S Ÿ AutpSq. ThenQ ď P1. Since the actions of P and P1 on S are equivalent, we have thatCNSpQqQpP1q “ 1. By Lemma 3.2, we have that P1 is a Sylow p-subgroup

of A. Since P1 ď NApQq, we have that P1 “ pQxϕyqt for some t P NSpQq

(proceed as in Step 2 of the proof of Lemma 3.2). Hence Q ď P2 “ pP1qt´1“


42 3.2. Character correspondences associated with PSL2p33aq

Qxϕy. We notice that

CNSpQqQpP2q “ pCNSpQqQpP1qqt´1“ 1,

IrrxϕypQq “ IrrP2pQq “ IrrP1pQq and IrrxϕypSq “ IrrP pSq. This proves theclaim.

Step 3. We describe the set IrrxϕypSq.We have that q ” 3 mod 4, for a ě 1. The character table of S has been

given in Section 1.5. We keep the notation of Section 1.5 throughout thisproof. Also in Section 1.5, it is is shown that t1S , StS , η

10, η

20u Ď IrrxϕypSq.

By Lemma 15.1 of [IMN07], we have that IrrxϕypSq “ t1S ,StS , η10, η

20u. l

The only character of this set with degree divisible by p is the Steinbergcharacter StS . Of course p1SqQ “ 1Q has the desired form. Hence, in orderto prove the statement, we need to understand how does α P tη10, η

20u restrict

to Q.

Step 4. We compute pη10qQ and pη20qQ.We denote by Tr the trace map TrF F3

: F Ñ F3 associated to the field

extension F F3. The trace map is F3-linear and Trpbq “ b` bp` ¨ ¨ ¨ ` bpa´1

for every b P F , by Corollary 23.11 of [Isa94]. Fix ε a primitive cubic root ofunity. For every b P F , we define a homomorphism λb : QÑ Cˆ as follows:

if c P F , then λb

ˆ

1 c0 1

˙

“ εTrpbcq. We have that

IrrpQq “ tλb | b P F u.

The automorphism ϕ acts on Q. Hence ϕ acts on IrrpQq. In fact λϕb “ λϕpbqfor every b P F . By Theorem 1.16, the number of elements of Q fixed by ϕis equal to the number of irreducible characters of Q fixed by ϕ. Since thesubfield of F fixed by ϕ is the prime field F3, we have that ϕ fixes exactlythree elements of Q. It is clear that IrrxϕypQq “ t1Q, λ1, λ´1u.

We denote by U the subgroup of Fˆ consisting of the squares of Fˆ.The subgroup U has index 2 in Fˆ. Since ´1 is not a square in F , everyb P Fˆ is either in U or there exists an u P U such that b “ ú. From thecharacter table of S we see that

pη10qQ ` pη20qQ “ ρQ ´ 1Q,

where ρQ is the regular character of Q (we recall ρQ has degree |Q| “ q andρQ vanishes on every nontrivial element of Q). Thus

ρQ “ÿ

bPF

λb “ 1Q ` pη10qQ ` pη

20qQ.

We conclude that pη10qQ is the sum of 12pq ´ 1q nontrivial λb’s and pη20qQ is

exactly the sum of the 12pq ´ 1q nontrivial λb’s that do not appear in pη10qQ.

The normalizer of Q in S is

NSpQq “ t

ˆ

c b0 c´1

˙

| c P Fˆ, b P F u “ď

cPFˆ

ˆ

c 00 c´1

˙

Q.



Let b P F . Write dpcq “ diagpc, c´1q P H, where c P Fˆ. It is straightforwardto check that

λdpcqb “ λc2b.

Thus the action of NSpQq on IrrpQq decomposes IrrpQq into three orbits asfollows

IrrpQq “ t1Qu Y tλb | b P Uu Y tλ´b | b P Uu.

Hence, if λb is a consituent of pη10qQ, then

pη10qQ “ÿ

uPU

λub

is exactly the sum over all squares of F if b is a square or the sum overall the non-squares of F if b is a non-square. In particular, we have thatλe is a constituent of pη10qQ if and only if λé is a constituent of pη20qQ fore P t´1, 1u. This finishes the proof of the statement.

Remark 3.5. Under the hypothesis of Lemma 3.2, we have that

IrrP pSq “ t1S ,StS , η1, η2u,

where StS is the Steinberg character of S and η1 and η2 are the two ir-reducible cuspidal characters of S of degree 1

2pq ´ 1q, by Lemma 15.1 of[IMN07].

3.3. An extension theorem

In this section we prove a non-trivial extension result which is key to provingTheorem E.

Theorem 3.6. Let N Ÿ G and let p be an odd prime. Let P P SylppGqand suppose that NGpP q “ P . Let χ P Irrp1pGq. If θ P IrrpNq lies under χ,then θ extends to Gθ.

The proof of Theorem 3.6 we present here is totally different from theone we originally gave in [NTV14]. This new shorter and cleaner proof isbased upon the ideas contained in [NT16]. In [NT16] the authors wereinterested in the case p “ 2, but their argument becomes easier for oddprimes.

We begin with an elementary extension result.

Lemma 3.7. Let N be a normal p-subgroup of G. Let χ P Irrp1pGq andlet θ P IrrpNq be under χ. Then θ extends to Gθ.

Proof. We may assume that θ is G-invariant since the Clifford corre-spondent ψ P IrrpGθ|θq of χ has p1-degree. Let P P SylppGq. Then N ď P .Since χ has p1-degree, some irreducible consituent µ of χP has p1-degree. Inparticular, µ is linear and lies over θ, hence µN “ θ. If Q P SylqpGq forsome prime q ‰ p, we have that θ also extends to NQ by Corollary 6.20 of[Isa76]. According to Corollary 11.31 of [Isa76] θ extends to G.


44 3.3. An extension theorem

We shall also need an extension result from [NT16], which is not hardto prove.

Lemma 3.8. Let N Ÿ G. Suppose that N “ S1 ˆ ¨ ¨ ¨ ˆ St is the directproduct of subgroups which are transitively permuted by G by conjugation.Write S “ S1 and view SZpSqŸ A “ AutpSq. Let θ “ θ1ˆ ..ˆ θt P IrrpNqbe G-invariant, where θi P IrrpSiq and θ1 P IrrpSZpSqq. If θ1 extends toAθ1, then θ extends to G.

Proof. This is Lemma 2.8 of [NT16].

We are ready to prove the main result of this section.

Proof of Theorem 3.6. Choose pG,Nq a counterexample of minimal|G| ` |N |. We may assume that G “ Gθ by minimality of pG,Nq since theClifford correspondent ψ P IrrpGθ|θq of χ has p1-degree and |G : Gθ| is ap1-number. Thus χN “ eθ for some e ě 1.

Suppose M Ÿ G and M ă N . Then

χM “ ftÿ

i“1

τxi ,

where τ P IrrpMq and tτx1 , ..., τxtu is a G-orbit. Also

χM “ eθM “ ef 1rÿ

i“1

τni ,

where the sum now is over an N -orbit. Write I “ Gτ . Then

|G : I| “ t “ r “ |N : N X I|,

so G “ NI. By Lemma 2.3 some irreducible constituent of χM is P -invariant, let us say τ , so P ď I. Let ρ P IrrpN X I|τq be the Clif-ford correspondent of θ. Since both τ and θ are I-invariant, also ρ is I-invariant. Now, let ψ P IrrpIq be under χ and over ρ. In particular, ψ liesover τ and then it must be the Clifford correspondent of χ over τ . HenceψG “ χ and ψ P Irrp1pIq. By minimality of pG,Nq, the character ρ ex-

tends to some µ P IrrpIq. Notice that I “ Gρ, so that µG P IrrpGq andpµGqN “ pµNXIq

N “ ρN “ θ, a contradiction.Hence, we may assume that N is a minimal normal subgroup of G. If N

is a p-group, then Lemma 3.7 yields a contradiction. If we suppose that N isa p1-group, then the hypothesis NGpP q “ P implies CN pP q “ 1 by coprimeaction. By the Glauberman correspondence (see Theorem 1.17) θ “ 1N ,which obviously extends to G, a contradiction. We may hence assume thatN “ S1 ˆ ¨ ¨ ¨ ˆ Sk is the direct product of some simple non-abelian groupstS1, ..., Sku of order divisible by p which are transitively permuted by G.Write θ “ θ1 ˆ ¨ ¨ ¨ ˆ θk P IrrpNq with θi P IrrpSiq. Write S “ S1 andSi “ Sxi for some xi P G for i “ 2, . . . , k. By Lemma 3.1, S is isomorphicto PSL2p3

3aq for some a ě 1. View S Ÿ A “ AutpSq. In Section 1.5.1 wehave seen that |AS| “ 2 ¨ 3a. Notice that opθ1q “ 1 because S1 has no



non-principal linear character. If QS P Syl3pAθ1Sq, then θ1 extends to Qby Lemma 1.13. If P S P Syl2pAθ1Sq it must be cyclic or trivial, in anycase, θ1 extends to P by Theorem 1.14. By Theorem 1.15, we conclude thatθ1 extends to Aθ. By Lemma 3.8, we see that θ extends to G, contradictingthe choice of G as a minimal counterexample.

3.4. The self-normalizing case

In this section we prove Theorem E. We also present a (perhaps surpris-ing) consequence of Theorem E: a characterization of groups having a self-normalizing Sylow p-subgroup in terms of the decomposition of a certainpermutation character (for odd primes). We shall need the following resultfrom [NTT07].

Lemma 3.9. Suppose that a finite p-group P acts on a finite group G,stabilizing N Ÿ G. Suppose that QN P SylppGNq is P -invariant, andassume that GN “ T1N ˆ ¨ ¨ ¨ ˆ TrN , where the Ti’s are permuted by P .Let Qi “ QXTi, and let Pi be the stabilizer of Ti in P . If CNGpQqQpP q “ 1,then CNTi

pQiqQipPiq “ 1.

Proof. Apply Lemma p4.1q of [NTT07] to each of the orbits definedby Q on tT1, . . . , Tru.

The only way we have found to prove Theorem E is to use a stronginduction over normal subgroups, and Theorem 3.6 is key in this inductiveprocess. The following result is a relative to normal subgroups version ofTheorem E.

Theorem 3.10. Let G be a finite group, p an odd prime, P P SylppGq,and suppose that P “ NGpP q. Let LŸ G. Let χ P Irrp1pGq. Then

χLP “ χ˚ `∆,

where χ˚ P Irrp1pLP q and either ∆ is zero, or ∆ is a character of LP whoseirreducible constituents have all degree divisible by p.

Proof. Let G be a counterexample with |G| ¨ |GL| smallest possible.

(a) By Lemma 2.3, let θ P IrrpLq be P -invariant under χ. Let T “ Gθ ěLP , and let ψ P IrrpT |θq be the Clifford correspondent of χ over θ. Assumethat T ă G. By the choice of G, we have that

ψLP “ ψ˚ `∆ ,

where ψ˚ has p1-degree and the irreducible constituents of ∆ have degreedivisible by p. Let T be a transversal for the double cosets of T and P inG. We may assume 1 P T Write

G “ď

xPTTxP


46 3.4. The self-normalizing case

Then, by Mackey’s Lemma 1.8, we have that

χLP “ pψGqLP “ ψLP `

ÿ

1‰xPTppψxqTxXLP q

LP .

Let α P IrrP pLq be an irreducible constituent of ppψxqTxXLP qLP . Suppose

that α has degree not divisible by p. Hence αL P IrrpLq. Thus the irreduciblecharacter αTxXLP lies under ψx. However pψxqL “ dθx for some d ě 1, sowe conclude that θx “ αL. Hence θx is P -invariant. By Lemma 2.3, we havethat θxy “ θ for some y P P and therefore x P T . But this is impossiblesince x ‰ 1 lies in T.

We may assume then that θ is G-invariant. By Theorem 3.6, we havethat θ has an extension θ P IrrpGq. By Gallagher Theorem 1.12, we have

that χ “ βθ, for some β P IrrpGLq. Now, if L ‰ 1, then the theorem holdsfor GL, whence we have that βPL is the sum of a p1-degree irreduciblecharacter β˚ of PLL (and hence linear) plus some character ∆ of PLLsuch that all of its irreducible constituents have degree divisible by p. Then

χLP “ pβ˚qθLP `∆θLP ,

and using Gallagher Theorem 1.12, we see that we are done again. HenceL “ 1.

(b) Suppose now that K is a minimal normal subgroup of G. Assumefirst that K is a p-group. Then |GK| ă |G| “ |GL|, and since KP “ P ,then the theorem is proved.

Assume next that K is a p1-group. Since NGpP q “ P , we have thatCKpP q “ 1. Since θ P IrrpKq is P -invariant, we necessarily have thatθ “ 1K by the Glauberman correspondence (see Theorem 1.17). But in thiscase, K ď kerpχq, and we can work in the group GK.

Hence, G has no abelian normal subgroup. In particular, F pGq “ 1.We have E “ F ˚pGq “ EpGq ě CGpEpGqq (see Theorem 6.5.8 of [KS04]).Since CGpEq “ ZpEq Ÿ G, it follows that ZpEq “ 1 and so E is product ofsubnormal nonabelian simple subgroups. By the main result of [GMN04],p “ 3 and G has a composition factor of type PSL2p3

3aq. By Lemma 3.1,E “ S1 ˆ . . . ˆ Sn is a direct product of non-abelian simple groups Si –PSL2pqiq, where qi “ 33ai with ai ě 1.

(c) Let Q “ P X E P SylppEq and write Q “ Q1 ˆ . . . ˆ Qn withQi P SylppSiq. Since P is self-normalizing in EP , by part (ii) of Lemma2.1 of [NTT07], we have that CNGpQqQpP q “ 1. This in turn implies,by Lemma 3.9, that CNSi

pQiqQipPiq “ 1 for Pi “ PSi . By Lemma 3.2, it

follows that Pi must induce the full subgroup C3ai of field automorphisms ofSi. By Remark 3.5, we see that the Pi-invariant irreducible characters of Siare αi “ 1Si , the Steinberg character StSi of degree qi and the two cuspidalcharacters η1i and η2i of degree 1

2pqi´1q. Furthermore for each α P tαi, η1i, η

2i u,

Pi fixes a unique irreducible constituent of αQi , appearing with multiplicityone, by Lemma 3.4. We denote by α˚ this constituent, so that the map ˚

defines a bijection from the Pi-invariant irreducible characters of p1-degree



of Si onto the Pi-invariant irreducible characters of Qi, again this is Lemma3.4.

Since the theorem holds for pG,Eq, χEP “ χ˚`∆, where χ˚ P Irrp1pEP qand all irreducible constituents of ∆ have degree divisible by p. In particular,θ “ pχ˚qE is irreducible. Write

θ “ θ1 ˆ ¨ ¨ ¨ ˆ θn

with θi P Irrp1pSiq. Since θ is P -invariant, it follows that θi is Pi-invariantof p1-degree, and so θi P tαi, η

1i, η

2i u. As mentioned above,

pθiqQi “ θ˚i ` δi,

where θ˚i P IrrpQiq is Pi-invariant, and δi is a sum of non-Pi-invariant irre-ducible characters of Qi. Setting

θ “ θ˚1 ˆ ¨ ¨ ¨ ˆ θ˚n,

we see that each irreducible constituent of θQ ´ θ is non-P -invariant and somust lie under an irreducible character of P of degree divisible by p. Butp does not divides χ˚p1q. Hence pχ˚qP contains a linear constituent which

must be unique and lies above θ. Denote this constituent by θ˚. We haveshown that every irreducible constituent of pχ˚qP ´ θ

˚ is of degree divisibleby p, whereas θ˚p1q “ 1.

(d) It remains to show that every irreducible constituent of ∆P hasdegree divisible by p. Assume the contrary: ∆P contains a linear constituentλ, and write

λQ “ λ1 ˆ ¨ ¨ ¨ ˆ λn

with λi P IrrpQiq. Let γ P IrrpEP q be an irreducible constituent of ∆ thatcontains λ upon restriction to P . Also, let

β “ β1 ˆ ¨ ¨ ¨ ˆ βn P IrrpEq

be lying under γ and above λQ. Since EG, we have that β is G-conjugateto θ. Note that the βi’s are Pi-invariant of degree not divisible by p, thusβi P tαi, η

1i, η

2i u. As shown in (c), the restriction pβiqQi contains a unique

Pi-invariant irreducible constituent β˚i , of multiplicity one. Denoting

β “ β˚1 ˆ ¨ ¨ ¨ ˆ β˚n,

we see that no irreducible constituent of βQ ´ β can be invariant under P .

But λQ lies under β and is P -invariant. Hence λQ “ β and λi “ β˚i . Nowwe consider two cases.

Case 1: β is not P -invariant. In this case, there is some g P P suchthat βg ‰ β. Then βg lies above pλQq

g “ λQ and under γ. Writing βg “β11 ˆ . . . ˆ β1n with β1i P tαi, η

1i, η

2i u for βg is G-conjugate to θ and so the β1i

are Pi-invariant. Arguing as above, we see that β˚i “ λi “ pβ1iq˚. By Lemma

3.4, the map α ÞÑ α˚ is a bijection. It follows that βi “ β1i and so β “ βg, acontradiction.


48 3.4. The self-normalizing case

Case 2: β is P -invariant. Then, by Theorem 3.6, β extends to β PIrrpEP q. Since γ lies above β and p divides γp1q, we have that γ “ βµ,where µ P IrrpP Qq is considered as a character of EP E and p dividesµp1q. Certainly, µλ is irreducible over P and non-linear. On the other hand,as shown above, every irreducible constituent of

pγP ´ µλqQ “ µQβQ ´ µQβ “ µp1q ¨ pβQ ´ βq

is non-P -invariant and so must lie under an irreducible character of P ofdegree divisible by p. Thus the degree of every irreducible constituent ofγP ´ µλ is divisible by p. Consequently, λ cannot be a constituent of γP , acontradiction.

Theorem E is now a corollary of Theorem 3.10.

Corollary 3.11. Let G be a finite group, let p be an odd prime and letP P SylppGq. Suppose that P “ NGpP q. If χ P Irrp1pGq, then

χP “ χ˚ `∆ ,

where χ˚ P IrrpP q is linear and ∆ is either zero or ∆ is a character whoseirrreducible constituents have all degree divisible by p. Furthemore, the mapχ ÞÑ χ˚ is a natural bijection Irrp1pGq Ñ IrrpP P 1q.

Proof. The first part follows from considering L “ 1 in Theorem 3.10.Let λ P IrrpP P 1q. Then

λG “ a1χ1 ` ¨ ¨ ¨ ` anχn,

where the ai’s are natural numbers and χi P IrrpGq. Since λGp1q “ |G : P |,some χi must have degree not divisible by p. So χi P Irrp1pGq. By Theorem3.10, pχiqP “ pχiq

˚`∆, where pχiq˚ is linear and no irreducible constituent

of ∆ is linear. However, λ is a linear constituent of pχiqP , and so pχiq˚ “ λ.

Thus, ˚ is surjective. By the main theorem of [GMN04] and Theorem Aof [IMN07], we have that |Irrp1pGq| “ |IrrpP P

1q|. Hence, the natural map˚ defines a bijection.

We finish this section with an application of Theorem E. We obtainthe following characterization of groups having a self-normalizing Sylow p-subgroup, for an odd prime p.

Corollary 3.12. Let G be a finite group, let p be odd, and let P P

SylppGq. Then NGpP q “ P if and only if

p1P qG “ 1G ` Ξ ,

where Ξ is either zero or a character whose irreducible constituents all havedegree divisible by p.

Proof. One implication follows from Corollary 3.11. Assume now that

p1P qG “ 1G ` Ξ ,



where Ξ is either zero or a character whose irreducible constituents all havedegree divisible by p, but N “ NGpP q ą P . Then there exists a non-principal character γ P IrrpNP q, which can be viewed as a character ofN . Since γ has p1-degree (because NP is a p1-group), it follows that γG

possesses an irreducible constituent χ P Irrp1pGq. Now, χ lies over γ ‰ 1Nand therefore χ ‰ 1G lies over 1P , a contradiction.

It is remarkable that Corollary 3.12 gives the exact opposite of a resultby G. Malle and G. Navarro in [MN12]: a finite group G has a normalSylow p-subgroup if and only if all irreducible constituents of p1P q

G havedegree not divisible by p.

The conclusion of corollary 3.12 is false for p “ 2, as shown by G “ S5:in this case p1P q

G contains the trivial character of G and an irreduciblecharacter of degree 5.

3.5. The p-decomposable Sylow normalizer case

Let N be a group and let p be a prime. We say that N is p-decomposableif N “ P ˆ X, where P P SylppNq. Suppose that the group G has a p-decomposable Sylow p-normalizer for an odd prime p (by Schur-Zassenhaus’theorem [Isa08, Thm. 3.5] this is equivalent to NGpP q “ CGpP qP forP P SylppGq). Then we can prove the following.

Theorem 3.13. Let G be a finite group, let p be an odd prime, and letP P SylppGq. Suppose that NGpP q “ PCGpP q. If χ P Irrp1pGq lies in theprincipal block, then

χNGpP q “ χ˚ `∆ ,

where χ˚ P IrrpNGpP qq is linear in the principal block, and ∆ is either zeroor ∆ is a character whose irreducible constituents all have degree divisibleby p. Furthermore, the map χ ÞÑ χ˚ is a bijection

Irrp1pB0pGqq Ñ Irrp1pB0pNGpP qqq,

where Irrp1pB0pGqq is the set of irreducible characters in the principal blockof G of degree not divisible by p.

Theorem 3.13 is the main result of this section. We begin by provingTheorem 3.14 below. Theorem 3.14 extends a classical result by J. Thomp-son (see Theorem 3.14 of [Isa08]) and it will be key for us later in thissection.

Theorem 3.14. Let G be a group, let p be a prime, and let P P SylppGq.Suppose that NGpP q “ P ˆ X. If p is odd or G is p-solvable, then X ď

Op1pGq. In particular, if p is odd or G is p-solvable and also NGpP q “PCGpP q, then Op1pNGpP qq ď Op1pGq.

Proof. We argue by induction on |G|. If LŸ G, then

NGLpPLLq “ NGpP qLL “ PLLˆXLL .


50 3.5. The p-decomposable Sylow normalizer case

Hence, if L ą 1, then we have that XLL ď Op1pGLq. In particular, wemay assume that Op1pGq “ 1. Now, suppose that N “ OppGq ą 1. ThenXNN ď Op1pGNq implies that X ď Opp1pGq “ M . Since rX,P s “ 1,then rX,N s “ 1. By Schur-Zassenhaus theorem [Isa08, Thm. 3.5], N hasa complement in H in M , similarly ZpNq has a complement K in CM pNqwhich must be contained in some conjugate Hg of H. Since K Ÿ Hg itfollows that K Ÿ M , so K Ď Op1pMq. However, using that Op1pGq “ 1, wehave that CM pNq “ ZpNqÔp1pMq “ ZpNq, and we conclude that X “ 1,in this case. Hence, we may assume that G is not p-solvable, and that p isodd.

Now, let N be a minimal normal subgroup of G. By [GMN04], wehave that N “ S1 ˆ ¨ ¨ ¨ ˆ Sk, where tS1, . . . , Sku are transitively permutedby G and S1 “ S “ PSL2p3

3aq. Now P X N “ pP X S1q ˆ ¨ ¨ ¨ ˆ pP X Skq.Fix some index i. Since rP,Xs “ 1, then rQi, Xs “ 1, where 1 ă Qi “P X Si P Syl3pSiq. Now, if x P X, then we have that pSiq

x “ Sj for some j.However Qxi ď Sxi X Si “ Sj X Si, so we conclude that X ď NGpSiq for alli with rX,Qis “ 1. Let Yi “ XCGpSiqCGpSiq, which is a 31-subgroup ofAutpSiq centralizing the Sylow 3-subgroup Qi of Si. By Lemma 1.5, Yi “ 1,whence X ď CGpSiq for all i. Thus X ď CGpNq for every minimal normalsubgroup. Since FpGq “ 1, then F˚pGq “ EpGq “ E. Since ZpEq “ 1, thenE is semisimple and CGpEq “ 1 by Theorems 9.7 and 9.8 of [Isa08]. Now Eis a direct product of non-abelian simple groups Ki and the normal closureof Ki is a minimal normal subgroup of G (by Lemma 9.17 of [Isa08], forinstance), and we conclude that X ď CGpEq “ 1, as desired.

Finally, since CGpP q “ ZpP q Ôp1pNGpP qq (by the Schur-Zassenhaustheorem [Isa08, Thm. 3.5]), it follows that if NGpP q “ PCGpP q, then

NGpP q “ P Ôp1pNGpP qq,

and we apply the first part of the theorem.

Note that the conclusion of Theorem 3.14 is not true for p “ 2: IfG “ E6p11q and P P Syl2pGq, then NGpP q “ P ˆC5, cf. [KM03, Theorem6(c)].

Let G be a group and let p be a prime. We denote by G0 the set ofp-regular elements of G (those elements whose order is not divisible by p).For the purpose of this exposition, we define the irreducible characters inthe principal (p)-block of G as the set of χ P IrrpGq satisfying

ÿ

xPG0

χpxq ‰ 0.

We write IrrpB0pGqq to denote the set of the irreducible characters of G thatlie in the principal block of G. (See Theorem 3.19 of [Nav98]).

Theorem 3.15. Let N “ Op1pGq. Suppose that χ P IrrpGq lies in theprincipal block of G, then N Ď kerpχq and the corresponding χ P IrrpGNqlies in the principal block of GN .



Proof. We notice that x P G is p-regular if and only if Nx P GN isp-regular. Hence, we can write

G0 “ Nx1 Y . . .YNxt

as a disjoint union (the p-regular classes of G in N correspond to the classof N1 in GN). Let X be a representation afffording χ. We can extend X bylinearity to a representation CGÑ Matpn,Cq which we denote again by X.We also denote by χ the trace CG Ñ C of this representation. Notice thatχpř

gPG aggq “ř

gPG agχpgq for everyř

gPG agg P CG. If X Ď G, then we

write X “ř

xPX x P CG. By hypothesis, we have that χpG0q ‰ 0. Hence,

the matrix XpG0q “ XpNqXpx1q`¨ ¨ ¨`XpNqXpxtq is non-zero. In particular,

XpNq ‰ 0. Notice that for every g P G

XpNqXpgq “ XpNgq “ XpgNq “ XpgqXpNq,

so that XpNq is scalar. We conclude that the trace of XpNq is non-zero, thisis, 0 ‰

ř

nPN χpnq “ |N |rχN , 1N s. Consequently N Ď kerpχq. The followingobservation

ÿ

xPG0

χpxq “ |N |tÿ

j“1

χpxjq “ |N |tÿ

j“1

χpNxjq “ |N |ÿ

xPpGNq0

χpxq

proves the second statement of the theorem.

The proof of the following lemma is a straightforward consequence ofTheorem 3.15.

Lemma 3.16. Suppose that N is a normal subgroup of H, with N ď

Op1pHq. Suppose that H “ NU for some U ď H. Then restriction de-fines a bijection between the characters in the principal block of H and thecharacters in the principal block of U .

As we prove below, in the case where NGpP q “ P , every character ofp1-degree of G lies in the principal block.

Lemma 3.17. Let G be a group and let p be a prime. Suppose thatNGpP q “ P where P P SylppGq. Let χ P IrrpGq. Then

ÿ

xPG0

χpxq ” χp1q mod p.

In particular, if χ P IrrpGq has p1-degree, then χ lies in the principal blockB0pGq of G.

Proof. Let P act on G0 by conjugation. The set of fixed points underthis action is G0 X CGpP q “ 1 by the assumption NGpP q “ P . Since theorbits corresponding to non-trivial elements of G0 have length divisible byp we have that

ÿ

xPG0

χpxq ” χp1q mod p.


52 3.6. The p-solvable case

The second statement follows from the definition of IrrpB0pGqq.

We can finally prove Theorem 3.13. The key to do that is to reduce theproof to the self-normalizing case and then apply Theorem E. This reductionis possible fundamentally thanks to Theorem 3.14.

Proof of Theorem 3.13. Let G be a counterexample to the first partof the theorem with |G| as small as possible. Let N “ Op1pGq. UsingTheorem 3.14 we can write NGpP q “ PˆX, where X ď N . Write G “ GNand use the bar convention. Hence P “ PNN P SylppGq and NGpP q “

P ˆ X “ P , by elementary group theory. By Lemma 3.15, N Ď kerpχq. IfN ą 1, then considering χ as a character of G, by inductive hypothesis wehave that

χNGpP q“ χP “ χ˚ `∆,

where the character χ˚ is an irreducible character of p1-degree lying in theprincipal block of P “ PNN and ∆ is either zero or a character of PNNsuch that every irreducible constituent of ∆ has degree divisible by p. Now,Lemma 3.16 applies and we are done in this case. Hence, we may assumeN “ 1. In particular, NGpP q “ P and the first part of the statement followsfrom Theorem 3.10.

Now, we prove that our map χ ÞÑ χ˚ is a bijection. By Theorem 3.14we have that Op1pNGpP qq ď Op1pGq and by Theorem 3.15 we have thatOp1pGq is contained in the kernel of every χ P IrrpB0pGqq. Modding out byOp1pGq, we may assume that Op1pGq “ 1 and so NGpP q “ P . By Lemma3.17 every irreducible character of p1-degree of G lies in IrrpB0pGqq. We havethat Irrp1pB0pGqq “ Irrp1pGq and Irrp1pB0pP qq “ Irrp1pP q. By Lemma 3.1 all

the non-abelian composition factors of G are of type PSL2p33aq with a ě 1.

We know by Theorem A of [IMN07] that the McKay conjecture holds forG. Hence |Irrp1pGq| “ |Irrp1pP q| “ |IrrpP P

1q|. Now, if λ P IrrpP q is linear,

then some irreducible constituent χ of λG has p1-degree. Now, χP containsλ and by the first part of the proof it must be χ˚ “ λ. Our map χ ÞÑ χ˚ issurjective, and therefore injective.

It is entirely possible that, under the hypotheses of Theorem 3.13, anatural correspondence exists between all the characters in Irrp1pGq andIrrp1pNGpP qq (not only the characters in the principal blocks). We havebeen able to find it for p-solvable groups, see coming Section 3.6.

3.6. The p-solvable case

We finish this chapter by proving Theorem F. Our correspondence in Theo-rem F extends the Glauberman correspondence (see Theorem 1.17) and alsothe correspondence in Theorem 3.13. We shall use Bπ-theory, for which werefer the reader to Section 1.4.



Lemma 3.18. Suppose that L Ÿ G, P P SylppGq and NGLpPLLq “PLL. Assume that GL is p-solvable. Let θ P IrrpLq be P -invariant and

p1-special. Then there exists a unique θ P IrrpG|θq such that θ is p1-special.

Proof. We argue by induction on |G : L|. Let KL be a chief factor ofG, and notice that GK has self-normalizing Sylow p-subgroups, by elemen-tary group theory. Assume first that KL is a p-group, and let η P IrrpK|θqbe the unique p1-special character lying over θ, by using part (b) of Propo-sition 1.21. By uniqueness, η is P -invariant, and by induction, there is aunique p1-special character η P IrrpGq that lies over η (and therefore over θ).Now, if γ is any other p1-special character of G lying over θ and ψ P IrrpKqlies under γ and over θ, we have that ψ is p1-special by part (a) of Proposi-tion 1.20, and therefore ψ “ η, by uniqueness. But in this case, γ “ η, byusing the inductive hypothesis.

Suppose finally that KL is a p1-group. Then CKLpPLLq “ 1 us-ing that PLL is self-normalizing and coprime action. Hence, by Problem(13.10) of [Isa76], there exists a unique P -invariant τ P IrrpK|θq. Also, τis p1-special by part (a) of Proposition 1.21. Since |G : K| ă |G : L|, byinduction there exists a unique p1-special character τ of G lying over τ (andtherefore over θ). Suppose now that γ P IrrpGq is any other p1-special char-acter lying over θ. By Lemma 2.3, let ϕ P IrrpKq be P -invariant under γ,and, by Theorem (13.27) of [Isa76], let ρ P IrrpLq be P -invariant under ϕ.Then ρ and θ are P -invariant lying under γ, so ρ “ θ by Lemma 2.3. Thenϕ “ τ by the uniqueness of τ , and hence γ “ τ by induction.

We translate the statement of Theorem 1.26 for p-solvable groups.

Remark 3.19. Let G be a p-solvable group and let P P SylppGq. Forχ P IrrpGq, the following are equivalent:

(a) χ is a satellite of some ψ P BppGq of p1-degree.(b) There exists a linear character λ of P and a p1-special character

α P IrrpW q, where W is the maximal subgroup of G to which λ

extends, such that χ “ pαλqG, where λ is the unique extension of λto W with p-power order.

Proof. Assume that χ is a satellite of ψ P BppGq of degree not divisibleby p. Then there exists a nucleus pW,γq for ψ and a p1-special characterα P IrrpW q such that χ “ pαγqG. Notice that χp1q “ |G : W |αp1qγp1q isa p1-number. Hence, we have that γp1q “ 1 and W contain a full Sylowp-subgroup of G. We may assume P ď W , by conjugating the pairs pW,γqand pW,αq with an element of G. Let λ “ γP P IrrpP q. Then γ is theunique extension of λ to W with p-power order. The fact that γG P IrrpGqguarantees that W is maximal with the property that λ extends to W .

To prove the converse, just notice that pλqG P BppGq by Theorem 2.2 of[IN01].


54 3.6. The p-solvable case

Recall that whenever a group A acts on the irreducible characters IrrpGqof a group G, we write IrrApGq to denote the set of fixed characters of Gunder the action of A. We can prove Theorem F, which we restate here.

Theorem 3.20. Let G be a p-solvable group, and let P P SylppGq.Suppose that NGpP q “ PCGpP q, and let L “ Op1pGq. Then for everyθ P IrrP pLq and λ P IrrpP P 1q linear, there is a canonically defined

λ ‹ θ P Irrp1pGq.

Furthermore, the map

IrrpP P 1q ˆ IrrP pLq Ñ Irrp1pGq

given by pλ, θq ÞÑ λ ‹ θ is a bijection. As a consequence, if θ˚ P IrrpCLpP qqis the Glauberman correspondent of θ P IrrP pLq (see Theorem 1.17), thenthe map

λˆ θ˚ ÞÑ λ ‹ θ

is a natural bijection Irrp1pNGpP qq Ñ Irrp1pGq. Also, if θ “ 1L and λ PIrrpP P 1q, then λˆ θ˚ is the unique linear constituent of pλ ‹ θqNGpP q.

Proof. By using Theorem 3.14, write NGpP q “ P ˆ X, where X “

CLpP q. Let λ P IrrpP q be linear and let θ P IrrP pLq. Since P X L “ 1,we trivially have that λ extends to PL. By Theorem 2.11, there exists amaximal subgroup P ďW ď G such that λ extends to W . Hence PL ďW .Now, by elementary character theory, let λ P IrrpW q be the unique linearcharacter of p-power order that extends λ. Since NW LpPLLq “ PLL, by

Lemma 3.18, there exists a unique p1-special θ P IrrpW q lying over θ. By

Theorem 2.2 of [IN01], the pair pW, λq is a nucleus for pλqG P IrrpGq. Thus,by Theorem 1.26 we have that

λ ‹ θ “ pθλqG P IrrpGq .

Notice that λ ‹ θ has p1-degree, because θ has p1-degree and |G : W | is not

divisible by p. (We notice for the record that pλ ‹ θqW contains θλ, andtherefore, when restricted to L, we have that pλ ‹ θq lies over θ. It is not ingeneral true that λ ‹ θ lies over λ, on the other hand.)

We have now defined a map

IrrpP P 1q ˆ IrrP pLq Ñ Irrp1pGq

given by pλ, θq ÞÑ λ ‹ θ.Next we show that our map is surjective. Let χ P Irrp1pGq. By Theorem

1.27 (see also Remark 3.19), we have that there exist a linear characterδ P IrrpP q and a p1-special character α P IrrpUq, where U is the maximalsubgroup of G to which δ extends, such that

χ “ pδαqG ,

where the order of δ is a p-power and δ extends δ. Now, αL contains a(unique) P -invariant character µ P IrrP pLq by Lemma 2.3, and it follows



that α is the unique p1-special character of U lying over µ by Lemma 3.18.It follows then that χ “ δ ‹ µ, and therefore, that our map is surjective.

Recall that the Glauberman correspondence provides a natural bijection

IrrP pLq Ñ IrrpCLpP qq .

Since the McKay conjecture is true for p-solvable groups (see for instance[IMN07]) we have that

|Irrp1pGq| “ |Irrp1pNGpP qq| “ |IrrpP P1q||IrrpCLpP qq| “ |IrrpP P

1q||IrrP pLq| .

It then follows that our map is bijective.In the case where θ “ 1L, for every λ P IrrpP P 1q, we have that λ ‹ θ “

pλqG, where λ is a p-special extension of λ to a subgroup W ď G with theproperty that λ does not extend to any subgroup of G properly containingW . Let T be a set of representatives of the double cosets of NGpP q and Win G with 1 P T. By Mackey Lemma 1.8, we have that

pλGqNGpP q “ λˆ 1`ÿ

1‰tPTppλtqNGpP qXW tqNGpP q.

By the first part of the proof ppλtqNGpP qXW tqNGpP q P IrrpNGpP qq, and soλˆ 1 is the unique linear constituent of pλ ‹ θqNGpP q.


CHAPTER 4

Preliminaries on modular character theory offinite groups

The first part of this chapter is the modular version of Chapter 1. Our aimis to collect the basic results on (p-)Brauer characters that will be used laterin this work, namely in Chapter 5. However, since the results containedin Chapter 5 are of a more technical nature (than the rest of results ofthis work), in the second part of this chapter we shall develop some newtechniques on Brauer characters.

In Section 4.1, we introduce Brauer characters and we focus on propertiesthat Brauer characters share with ordinary characters. In Section 4.2, westudy a modular version of the notion of central isomorphism of charactertriples firstly defined in [NS14] and later developed in [Spa16]. In Section4.3, we introduce the concept of fake Galois action on the set of Brauerirreducible characters IBrpNq of a group N . This latter definition will helpus to avoid some difficulties in Chapter 5 originated from the fact that, ingeneral, the Galois group GalpQ|N |Qq does not act on IBrpNq. (See Section5.4 for the motivation of the definition of fake Galois action.)

4.1. Brauer characters (as characters)

We fix a prime p and a maximal ideal M in the ring R of algebraic integerswith p PM. We let F “ RM and write ˚ : R Ñ F to denote the naturalring homomorphism. This homomorphism can be extended to S “ trs | r PR, s P RzMu by

prsq˚ “ r˚ps˚q´1,

for every r P R and s P RzM. Let U Ď R be the multiplicative groupof roots of unity of order not divisible by p, so that U “ tξ P C | ξk “1 for some integer k not divisible by p u.

Lemma 4.1. The restriction of ˚ to U defines an isomorphism U Ñ Fˆ

of multiplicative groups. Also F is an algebraically closed field of character-istic p.

Proof. This is Lemma 2.1 of [Nav98].

Let g P G. We say that x is p-regular if p does not divide opgq. Werecall that G0 denotes the subset of p-regular elements of G. Suppose thatX : G Ñ GLnpF q is an F -representation of the group G. If g P G0, thenby Lemma 4.1, the eigenvalues of the matrix Xpgq are ξ˚1 , . . . , ξ

˚n P F

ˆ for

57

58 4.1. Brauer characters (as characters)

uniquely determined ξ1, . . . , ξn P U (because F is algebraically closed). Wedefine ϕ : G0 Ñ C by ϕpgq “ ξ1 ` ¨ ¨ ¨ ` ξn. Then we say that ϕ is theBrauer character afforded by X. (Brauer characters are also called mod-ular characters.) The degree of ϕ is ϕp1q, which is the degree of anyF -representation affording ϕ. Notice that similar F -representations affordthe same Brauer characters and Brauer characters are constant on conjugacyclasses.

We say that ϕ is irreducible if an F -representation X of G affording ϕis irreducible. We denote by IBrpGq the set of irreducible Brauer charactersof G. Unlike ordinary characters, the degrees of the irreducible Brauercharacters do not divide, in general, the order of the group (PSL2p7q forp “ 7 has an irreducible Brauer character of degree 5).

We define the field of values Qpϕq of ϕ P IBrpGq as Qpϕpgq | g P G0q.Notice that Qpϕq Ď Q|G|p1 Ď Q|G|.

Write cfpG0q to denote the C-vector space of class functions on G0

(functions θ : G0 Ñ C constant on conjugacy classes of G0). Of course thedimension of cfpG0q is equal to the number of conjugacy classes of p-regularelements of G. Brauer characters are class functions on G0.

If H ď G and ϕ is a Brauer character of G, then we denote by ϕH therestriction of ϕ to H0. The map ϕH is a Brauer character of H.

As happens with ordinary characters, Brauer characters are nonnegativeinteger linear combination of irreducible Brauer characters.

Theorem 4.2. Let G be a group. Then IBrpGq is a basis of cfpG0q.Moreover, ψ P cfpG0q is a Brauer character of G if and only if

ψ “ÿ

ϕPIBrpGq

aϕϕ,

where aϕ P N.

Proof. See Corollary 2.10 and Theorem 2.3 of [Nav98].

The nonnegative integer aϕ in the decomposition of ψ in Theorem 4.2 iscalled the multiplicity of ϕ in ψ. If aϕ ‰ 0, then we call ϕ a constituentof ψ.

By Theorem 4.2, the number |IBrpGq| equals the number of conjugacyclasses of p-regular elements of G. Also, a Brauer character ϕ is irreducibleif and only if ϕ cannot be written as α` β for Brauer characters α and β.

As a consequence of Theorem 4.2, if ϕ P IBrpGq, then any two irreduciblerepresentations affording ϕ are similar. Hence, ϕ uniquely determines an F -representation of G up to similarity.

If χ P IrrpGq, we denote by χ0 the restriction of χ to G0. By Corollary2.9 of [Nav98], χ0 is a Brauer character of G. Hence

χ0 “ÿ

ϕPIBrpGq

dχϕϕ,


4. Preliminaries on modular character theory of finite groups 59

for suitable nonnegative integers dχϕ. The nonnegative integers dχϕ in theabove decomposition are called the decomposition numbers of χ.

The study of Brauer characters is more interesting when p divides |G|because of the following.

Theorem 4.3. Let G be a group. If p does not divide |G|, then IBrpGq “IrrpGq.

Proof. This is Theorem 2.12 of [Nav98].

As in the ordinary case, if ϕ and θ are Brauer characters of G, then theproduct ϕθ defined by

ϕθpgq “ ϕpgqθpgq

for every g P G0 is a Brauer character of G (see Theorem 2.23 of [Nav98]).

Let ϕ P IBrpGq and let n “ ϕp1q. Since ϕ uniquely determines an irre-ducible F -representation X up to similarity, then kerpXq is uniquely deter-mined by ϕ. We can define the kernel of ϕ as kerpϕq “ tg P G | Xpgq “ In u.As in the ordinary case, if N Ÿ G, one can identify the irreducible Brauercharacters of G containing N in their kernel and the irreducible Brauer char-acters of the quotient group GN . Usually we identify these sets and wethink of IBrpGNq as a subset of IBrpGq.

We recall that OppGq is the maximal normal p-subgroup of G.

Lemma 4.4. Let G be a group. If ϕ P IBrpGq, then OppGq Ď kerpϕq. Inparticular, we can identify IBrpGOppGqq with IBrpGq.

Proof. See Lemma 2.32 of [Nav98].

Let ϕ P IBrpGq. The fact that ϕ uniquely determines up to similarityan F -representation X of G allows us to define the determinantal order opϕqof ϕ as in the ordinary case. The determinant detpXq : G Ñ Fˆ of X isa homomorphism. We write opϕq to denote the smallest positive integer ksuch that detpXqk “ 1.

As for ordinary characters, every isomorphism of groups α : G Ñ Hdefines a bijection between IBrpGq and IBrpHq, by defining for ϕ P IBrpGqthe map ϕα given by ϕαpgαq “ ϕpgq for every g P G0. In particular, ifN Ÿ G, then G acts on IBrpNq via the action of conjugation of G on N .

Induction and restriction are essential features also in modular charactertheory. We have already noticed that the restriction of a Brauer characterto a subgroup is again a Brauer character. Let H ď G and let θ P cfpH0q.We define for every x P G0, the function

θGpxq “1

|H|

ÿ

gPG

9θpgxg´1q,

where 9θpyq “ θpyq if y P H0 and 0 otherwise. Then θG P cfpG0q. In fact, wehave the following.


60 4.1. Brauer characters (as characters)

Theorem 4.5. Let H ď G and let ϕ be a Brauer character of H. ThenϕG is a Brauer character of G.


As for ordinary characters, ifH ď G and θ P IBrpHq, we write IBrpG|θq “tϕ P IBrpGq | ϕ is a constituent of θG u (this set is non-empty by Corollary8.3 of [Nav98]). If ϕ P IBrpG|θq we will say that ϕ lies over θ or that θlies under ϕ. If ϕ P IBrpGq, then we write IBrpϕHq to denote the set ofirreducible Brauer characters which are constituents of the Brauer characterϕH .

The main difference between ordinary and modular characters with re-spect to the induction-restriction process is that in the modular case we donot have Frobenius reciprocity. For H ď G, this means that if θ P IBrpHqand ϕ P IBrpGq, then the multiplicity of ϕ in θG is not necessarily equal tothe multiplicity of θ in ϕH . In fact, it can happen that ϕ is a constituent ofθG but θ is not a constituent of ϕH and viceversa (see the discussion afterCorollary 8.3 of [Nav98] for these extreme examples).

Despite this fact, restriction-induction techniques with respect to normalsubgroups work exactly as well as for ordinary characters.

Theorem 4.6. Let N Ÿ G. If θ P IBrpNq and ϕ P IBrpGq, then ϕ is aconstituent of θG iff θ is a constituent of ϕN . In this case,

ϕN “ etÿ

i“1

θxi ,

for some e ě 1, where x1 “ 1 and θx1 , . . . , θxt are the distinct G-conjugatesof θ.

Proof. This is Corollary 8.7 of [Nav98].

Let NŸG. If θ P IBrpNq, then we write Gθ “ tg P G | θg “ θ u to denote

the inertia group of θ in G. For H ď G, we say that θ is H-invariant ifH Ď Gθ.

Theorem 4.7 (Clifford correspondence). Let NŸ G and let θ P IBrpGq.Then the map ψ ÞÑ ψG is a bijection from IrrpGθ|θq onto IBrpG|θq.


In view of Theorem 4.6 and Theorem 4.7, we see that there is a Clif-ford theory for modular characters. Also, Gallagher’s Theorem works formodular characters.

Theorem 4.8. Let NŸ G and let ϕ P IBrpGq. If ϕN “ θ P IBrpNq, thenthe characters βϕ for β P IBrpGNq are irreducible, distinct for distinct βand they are all the irreducible constituents of θG.

Proof. See Corollary 8.10 of [Nav98].



Next, we collect some extendibility criteria. If H ď G and θ P IBrpHq,then we say that θ extends if there is ϕ P IBrpGq such that ϕH “ θ.

Theorem 4.9 (Green). Let N Ÿ G and let θ P IBrpNq. If GN is ap-group, then there exists a unique ϕ P IBrpG|θq. Furthermore, ϕN is thesum of the distinct G-conjugates of θ. In particular, if θ is G-invariant thenϕN “ θ.

Proof. See Theorem 8.11 of [Nav98].

The following criteria are modular analogues of well-known results onordinary characters.

Theorem 4.10. Let N Ÿ G. Suppose that GN is cyclic. If θ P IBrpNqis G-invariant, then θ extends to G.


Theorem 4.11. Let N Ÿ G and let θ P IBrpNq be G-invariant. If θextends to Q for every QN P SylqpGNq and for every prime q ‰ p, then θextends to G.


Theorem 4.12. Let N Ÿ G and let θ P IBrpNq be G-invariant. If oneof the following holds:

(a) p|N |, |G : N |q “ 1, or(b) popθqθp1q, |G : N |q “ 1,

then θ extends to G.

Proof. See Theorem 8.13 and Theorem 8.23 of [Nav98].

Notice that part (a) does not follow from part (b) since, as we said, thedegrees of irreducible Brauer characters do not divide in general the orderof the group.

4.2. Isomorphisms of modular character triples

In this section we start by introducing the theory of projective representa-tions we will later need. We will follow Chapter 8 of [Nav98] since we areinterested in the modular case, but this theory works exactly the same bothin the ordinary and the modular case (a reference for the ordinary case isChapter 11 of [Isa76]). After that, we will introduce the notion of centrallyisomorphic modular character triples, which is a modular analogue of thenotion of centrally isomorphic character triples introduced in [NS14] forordinary character triples. (Hence the proofs of most of the results concern-ing the construction of centrally isomorphic modular character triples willfollow from arguments contained in [NS14] and [Spa16].)

If NŸ G and θ P IBrpNq is G-invariant, then the triple pG,N, θq is calleda modular character triple. Every modular character triple pG,N, θq has


62 4.2. Isomorphisms of modular character triples

associated an (F -)projective representation P (up to similarity and up toproduct by some function µ : G Ñ Fˆ) satisfying certain properties. (SeeTheorem 4.13 and Remark 4.15 below.)

A projective representation P of G is a map P : GÑ GLnpF q satis-fying that for every g, g1 P G

PpgqPpg1q “ αpg, g1qPpgg1q,where αpg, g1q P Fˆ. We say that n is the degree of P. This definitionyields a function α : GˆGÑ Fˆ, which is the factor set (see page 164 of[Nav98]) associated to P.

The notions of similarity and irreducibility of projective representationsare analogous to those on representations.

Theorem 4.13. Let pG,N, θq be a modular character triple. Let X bean F -representation affording the Brauer character θ. Then, there exists aprojective representation P of G, such that PN “ X and the factor set αassociated to P satisfies

αpg, nq “ 1 “ αpn, gq,

for every g P G and n P N . Moreover, if Q is another such projectiverepresentation, then there exists a map µ : G Ñ Fˆ with µp1q “ 1 which isconstant on N -cosets in G and such that Q “ µP.


The factor set α associated to P in Theorem 4.13 can be seen as a mapdefined on GN ˆGN (see the remarks after Theorem 8.14 of [Nav98]).

Definition 4.14. Let pG,N, θq be a modular character triple. We saythat a projective representation P of G is associated with θ if:

(i) PN is a representation affording θ, and(ii) the factor set α of P satisfies

αpg, nq “ 1 “ αpn, gq

for every g P G and n P N .

Notice that if pG,N, θq is a modular character triple and P is a projectiverepresentation of G associated to θ, then condition (ii) in Definition 4.14implies that for every g P G and n P N

Ppgnq “ PpgqPpnq and Ppngq “ PpnqPpgq.Remark 4.15. Let pG,N, θq be a modular character triple. By Theorem

4.13, projective representations of G associated with θ do always exist. Infact, it is easy to prove that if P and Q are two projective representationsof G associated with θ, then there exists a map µ : G Ñ Fˆ with µp1q “ 1which is constant on cosets of N and such that Q is similar to µP. Let αbe the factor set of P. Then the factor set β of Q is given by

βpg, g1q “µpgqµpg1q

µpgg1qαpg, g1q



for every g, g1 P G. (This latter fact follows from straightforward calcula-tions.)

We can study the Clifford theory of a character triple pG,N, θq viaprojective representations associated to θ and projective representations ofGN . We will view the projective representations of GN as projectiverepresentations Q of G satisfying Qpgnq “ Qpgq for every g P G and n P N .

Let P be a projective representation associated with θ with factor setα (recall we can consider α as a function defined on GN ˆ GN). LetN ď J ď G and let γ denote the restriction of the factor set α´1 to JN ˆJN . By Theorem 8.16 and Theorem 8.18 of [Nav98], if ProjF pJN, γqis a set of representatives of the similarity classes of irreducible projectiveF -representations of JN with factor set γ, then

RepF pJ, θq “ tQb PJ | Q P ProjF pJN, γqu

is a set of representatives of similarity classes of representations affording aBrauer character in IBrpJ |θq.

Definition 4.16. Let pG,N, θq be a modular character triple. We de-note by BrpG|θq the set of Brauer characters χ of G such that χN is a mul-tiple of θ. Hence this is the set of nonnegative integer linear combinationsof IBrpG|θq. Let pΓ,M, ϕq be another modular character triple and supposethat τ : GN Ñ ΓM is an isomorphism of groups. For every N ď J ď G,write Jτ N “ τpJNq and suppose that there exists a map σJ : BrpJ |θq ÑBrpJτ |ϕq such that σJ yields a bijection IBrpJ |θq Ñ IBrpJτ |ϕq. Supposefurther that for every N ď K ď J ď G and for every χ, ψ P BrpJ |θq thefollowing hold:

(a) σJpχ` ψq “ σJpχq ` σJpψq.(b) σKpχKq “ σJpχqK .(c) σKpχβq “ σKpχqβ

τ , for every β P IBrpJNq.

Then we say that pσ, τq : pG,N, θq Ñ pΓ,M, ϕq is an isomorphism of mod-ular character triples.

To define an isomorphism pσ, τq of modular character triples it is enoughto give for N ď J ď G bijections

σJ : IBrpJ |θq Ñ IBrpJτ |ϕq,

extend these maps using (a) and check that conditions (b) and (c) hold forevery χ, ψ P IBrpJ |θq.

We strengthen Definition 4.16. Let N ď J ď G. For ψ P IBrpJ |θq andg “ gN P GN , we define ψg P IBrpJg|θq by

ψgpxgq “ ψpxq for every x P J.

Note that this is well-defined. We say that a modular character triple iso-morphism pσ, τq : pG,N, θq Ñ pΓ,M, ϕq is strong if

pσJpψqqτpgq “ σJgpψ

gq,



for all g P GN , all groups J with N ď J ď G and all ψ P IBrpJ |θq.

From now on, we will work with strong isomorphisms of character triples.Isomorphisms of modular character triples define an equivalence relation onmodular character triples.

We collect below some examples of (strong) modular character tripleisomorphisms.

Lemma 4.17. Let pG,N, θq be a modular character triple.

(a) If α : GÑ H is an isomorphism of groups, then pG,N, θq is stronglyisomorphic to pH,M,ϕq where M “ αpNq and ϕ P IBrpMq is thecharacter defined by ϕpαpnqq “ θpnq for every n P N0.

(b) If M Ÿ G and M ď kerpθq, then pG,N, θq and pGM,NM, θq arestrongly isomorphic, where θpnMq “ θpnq for every n P N0.

(c) Suppose that µ : GÑ H is an epimorphism and that K “ kerpµq ďkerpθq. Then pG,N, θq and pH,M,ϕq are strongly isomorphic, whereM “ µpNq and ϕ P IrrpMq is the unique character of M withϕpµpnqq “ θpnq for every n P N0.

(d) Suppose that there exists some η P IBrpGq such that ηNθ “ ϕ PIBrpNq. Then pG,N, θq and pG,N,ϕq are strongly isomorphic.

Proof. Parts (a) and (b) are particular cases of part (c). To prove(c) mimic the proof of Lemma 11.26 of [Isa76]. It is easy to check thatthe isomorphism obtained this way is strong. Part (d) is Theorem 8.26 of[Nav98]. As before, it is straightforward to check that the isomorphismgiven by this proof is strong.

The following result explains how to construct (strong) isomorphisms ofmodular character triples via projective representations.

Theorem 4.18. Let pG,N, θq and pH,M, θ1q be modular character triplessatisfying the following assumptions:

(i) G “ NH and M “ N XH,(ii) there exist projective representations P and P 1 of G and H associ-

ated to θ and θ1, respectively, whose factor sets α and α1 coincidevia the natural isomorphism τ : GN Ñ HM .

For N ď J ď G, write γ “ pα´1qJNˆJN . If ψ P IBrpJ |θq is affordedby Q b PJ , where Q P ProjF pJN, γq, then let σJpψq be the Brauer char-acter afforded by the irreducible representation Q b P 1JXH . Then the mapσJ : IBrpJ |θq Ñ IBrpJ X H|θ1q is a bijection. These bijections σJ togetherwith τ give a strong isomorphism

pσ, τq : pG,N, θq Ñ pH,M, θ1q

of modular character triples.

Proof. See the proof of Theorem 3.2 of [NS14]. The fact that pσ, τqis strong follows from straightforward calculations.



In the situation of Theorem 4.18, we say that pσ, τq is an isomorphismof modular character triples given by P and P 1.

We are finally ready to define when two modular character triples arecentrally isomorphic.

Definition 4.19. Let pG,N, θq and pH,M, θ1q be modular charactertriples satisfying the following conditions:

(i) G “ NH, M “ N XH and CGpNq ď H.(ii) There exist a projective representation P of G associated to θ with

factor set α and a projective representation P 1 of H associated toθ1 with factor set α1 such that

(ii.1) α|HˆH “ α1, and(ii.2) for every c P CGpNq the scalar matrices Ppcq and P 1pcq are

associated with the same scalar (notice that Ppcq and P 1pcqare scalar by Schur’s Lemma [Isa76, Lem. 1.5]).

Let pσ, τq be the isomorphism of character triples given by P and P 1 as inTheorem 4.18. Then we call pσ, τq a central isomorphism of modularcharacter triples, and we write

pG,N, θq ąBr,c pH,M, θ1q.

By the proof of Lemma 3.3 of [NS14], condition (ii.2) in Definition 4.19is equivalent to

IBrpψCJ pNqq “ IBrpσJpψqCJ pNqq,

for every ψ P IBrpJ |θq and N ď J ď G. Definition 4.19 is a modularanalogue of the relation „c defined in [NS14]. In particular, the fact thatąBr,c defines an order relation on the set of modular character triples andis thereby transitive follows from the proof of Lemma 3.8 of [NS14].

Remark 4.20. Suppose that pG,N, θq ąBr,c pH,M, θ1q. Then ZpNq ĎM . The condition (ii.2) in Definition 4.19 implies that θ and θ1 lie over thesame λ P IBrpZpNqq.

We are interested in studying how to construct new centrally isomorphicmodular character triples from given ones. In order to do that, we shallfrequently use the following.

Lemma 4.21. Let pG,N, θq and pH,M, θ1q be modular character tripleswith


(a) If P is any projective representation of G associated to θ with factorset α, then there exists a projective representation P 1 of H associ-ated to θ1 with factor set α1 such that

(a.1) α|HˆH “ α1, and(a.2) for every c P CGpNq the scalar matrices Ppcq and P 1pcq are

associated with the same scalar.



(b) If P 1 is any projective representation of H associated to θ1 withfactor set α1, then there exists a projective representation P of Gassociated to θ with factor set α such that

(b.1) α|HˆH “ α1, and(b.2) for every c P CGpNq the scalar matrices Ppcq and P 1pcq are

associated with the same scalar.

Proof. We first prove part (a). Let Q and Q1 be projective repre-sentations as in Definition 4.19 giving pG,N, θq ąBr,c pH,M, θ1q. In thesituation of (a), by Remark 4.15, there exists a map µ : G Ñ Fˆ and amatrix M P GLθp1qpF q such that P “M´1µQM . Let P 1 “ µQ1. Then P 1 is

a projective representation of H associated to θ1 whose factor set certainlysatisfies (a.2). By the second part of Remark 4.15, the factor set of P 1 alsosatisfies (a.1).

The proof of part (b) is analogous.

The following nearly trivial observation on centrally isomorphic modularcharacter triples will be useful later.

Lemma 4.22. Suppose that pG,N, θq ąBr,c pH,M, θ1q. Let Γ with N ď

G ď Γ and x P Γ. Then pGx, Nx, θxq ąBr,c pHx,Mx, pθ1qxq.

Proof. According to Definition 4.19 one can obtain projective represen-tations giving pGx, Nx, θxq ąBr,c pH

x,Mx, pθ1qxq, by using the isomorphismx : G Ñ Gx given by g ÞÑ gx, from those projective representations givingpG,N, θq ąBr,c pH,M, θ1q.

We analyze the behavior of centrally isomorphic modular character tripleswith respect to certain quotients.

Lemma 4.23. Suppose that pG,N, θq ąBr,c pH,M, θ1q. Let ε : GÑ G1 bean epimorphism. Write N1 “ εpNq, H1 “ εpHq and M1 “ εpMq. Supposethat Z “ kerpεq ď kerpθq X kerpθ1q and εpCGpNqZq “ CG1pN1q. Then

pG1, N1, θ1q ąBr,c pH1,M1, θ11q,

where θ1 P IBrpN1q is such that θ “ θ1 ˝ ε and θ11 P IBrpM1q is such thatθ1 “ θ11 ˝ ε.

Proof. See the proof of Corollary 4.5 of [NS14]. (Note that there thestronger assumption Z ď ZpGq in [NS14] is only used for the block-theoreticstatements that are not relevant in our context.)

Let Gi be finite groups for i “ 1, 2. Of course pG1 ˆ G2q0 “ G0

1 ˆ

G02. Recall that IBrpG1 ˆ G2q “ tθ1 ˆ θ2 | θi P IBrpGiqu. (See Theorem

8.21 of [Nav98].) The following lemma tells us how to construct centrallyisomorphic modular character triples using direct and semi-direct products.

If G is a group and m is an integer, then we denote by Gm the externaldirect product of m copies of G. The group Sm acts naturally on Gm bypg1, . . . , gmq

σ “ pgσ´1p1q, . . . , gσ´1pmqq.



Theorem 4.24. Suppose that pG,N, θiq ąBr,c pH,M, θ1iq, for 1 ď i ď m.Let θ “ θ1 ˆ ¨ ¨ ¨ ˆ θm P IrrpNmq, and θ1 “ θ11 ˆ ¨ ¨ ¨ ˆ θ

1m P IrrpHmq. Suppose

that A ď Sm stabilizes θ and θ1. Then

pGm ¸A,Nm, θq ąBr,c pHm ¸A,Mm, θ1q.

Proof. If σ P Sm, then θσ “ θσ´1p1q ˆ ¨ ¨ ¨ ˆ θσ´1pmq, (evaluate θσ onelements of the form p1, . . . , n, . . . , 1q for n P N). Therefore, σ fixes θ if andonly if θσpiq “ θi for every i P t1, . . . ,mu.

We may certainly assume that N ‰ 1. Notice that pn, 1, . . . , 1qσ “pn, 1, . . . , 1q for 1 ‰ n P N if and only if σp1q “ 1. Using this for everyi P t1, . . . ,mu, we check that CGmApN

mq “ CGpNqm Ď Hm.

Let Vi “ F θip1q for each i P t1, . . . ,mu. Write V “ V1b ¨ ¨ ¨bVm. Noticethat if σ P A, then Vi “ Vσpiq for every i P t1, . . . ,mu. Let σ P A. We define alinear map σ : V Ñ V by σpv1b¨ ¨ ¨bvmq “ vσ´1p1qb¨ ¨ ¨bvσ´1pmq for vi P Vi,and extending linearly to all tensors. This defines a group homomorphismAÑ GLpV q. Let Rpσq P GLnpF q be the matrix associated to σ, so that

RpσqpM1 b ¨ ¨ ¨ bMmqRpσq´1 “Mσp1q b ¨ ¨ ¨ bMσpmq

for matrices Mi P GLθip1qpF q.

By hypothesis, we have projective representations Pi and P 1i associatedwith θi and θ1i giving pG,N, θiq ąBr,c pH,M, θiq. Next, we define a projectiverepresentation P of Gm ¸A associated with θ. Set

Pppg1, . . . , gmqσq “ pP1pg1q b ¨ ¨ ¨ b PmpgmqqRpσq,

for gi P G and σ P A. It is easy to check that the factor set α of P satisfies

αppg1, . . . , gmqσ, pg11, . . . , g

1mqτq “

mź

i“1

αipgi, g1σpiqq,

for gi, g1i P G and σ, τ P A, where αi is the factor set of Pi for each

i P t1, . . . ,mu. Analogously, we construct a projective representation P 1of Hm ¸ A associated with θ1. Let α1i be the factor set of P 1i for eachi P t1, . . . ,mu. By construction, the factor set α1 of P 1 satisfies

α1pph1, . . . , hmqσ, ph11, . . . , h

1mqτq “

mź

i“1

α1iphi, h1σpiqq,

for hi, h1i P H and σ, τ P A. Since αi and α1i agree on H ˆH, we have that

α and α1 satisfy Definition 4.19(ii.1). Recall that CGm¸ApNmq “ CGpNq

m,hence P and P 1 trivially satisfy Definition 4.19(ii.2).

Let G “ G1 be a group, and let αi : G Ñ Gi be a group isomorphismfor each i P t1, . . . ,mu, with α1 “ idG. Let G “ G1 ˆ ¨ ¨ ¨ ˆ Gm. Then the

symmetric group Sm acts on G in the following way:

pxα11 , . . . , xαmm qσ “ ppxσ´1p1qq

α1 , . . . , pxσ´1pmqqαmq.



Corollary 4.25. With the previous notation, for each i P t1, . . . ,mu,assume that pGi, Ni, θiq ąBr,c pHi,Mi, θ

1iq, where Hi “ Hαi, Ni “ Nαi and

Mi “Mαi, for some subgroups N , M and H of G. Write H “ H1ˆ¨ ¨ ¨ˆHm,N “ N1 ˆ ¨ ¨ ¨ ˆ Nm, and M “ M1 ˆ ¨ ¨ ¨ ˆMm. Write θ “ θ1 ˆ ¨ ¨ ¨ ˆ θm,and θ1 “ θ11 ˆ ¨ ¨ ¨ ˆ θ

1m. Suppose pSmqθ “ pSmqθ1. Then

pG¸ pSmqθ, N , θq ąBr,c pH ¸ pSmqθ, M , θ1q.

Proof. Define α : Gm ¸ pSmqθ Ñ G ¸ pSmqθ by αppg1, . . . , gmqσq “

pgα11 , . . . , gαmm qσ. Then α is an isomorphism. Write β “ α´1, θ “ pθqβ,

θ1 “ pθ1qβ. By Theorem 4.24 we have that

pGm ¸ pSmqθ, Nm, θq ąBr,c pH

m ¸ pSmqθ,Mm, θ1q.

Let P and P 1 be projective representations giving the above central isomor-phism. Then it is easy to check that Pα and pP 1qα give the desired centralisomorphism of modular character triples.

The following key result is deeper than the others mentioned in thissection so far. It is basically a modular version of Theorem 5.3 of [Spa16]without taking into account p-blocks. The proof we present is the modularversion of the one given in [NS16].

Theorem 4.26. Let pG,N, θq ąBr,c pH,M, θ1q. Suppose that N Ÿ G and

GCGpNq is equal to GCGpNq as a subgroup of AutpNq. Let H ď G such

that H ě CGpNq, and HCGpNq and HCGpNq are equal as subgroups ofAutpNq. Then


Proof. By assumption we have that H “ tx P G | for some h P H, nx “nh for every n P Nu.

We start by proving that G “ NH. Let x P G. Then there exists someg P G such that mx “ mg for every m P N . Since g “ hn for some h P Hand n P N , we conclude xn´1 P H. Therefore G “ NH. Now, we prove thatN X H “M “ N XH. The inclusion N XH Ď N X H is obvious. Assumenow that m P N X H. Hence there is some h P H such that nm “ nh forevery n P N . Hence mh´1 P CGpNq Ď H. Thus n P H and n P NXH “M .

Therefore N X H ĎM .

The map τ : GCGpNq Ñ GCGpNq given by τpxCGpNqq “ yCGpNqif and only if x and y induce the same conjugation automorphism of N isa group isomorphism sending NCGpNqCGpNq onto NCGpNqCGpNq. Infact, τpnCGpNqq “ nCGpNq.

Notice that θ is G-invariant. This is derived from the fact that θ is G-invariant since every element of G acts on N as an element of G. Similarlyθ1 is H-invariant.



By assumption pG,N, θq ąBr,c pH,M, θ1q, so there exist projective rep-resentations P of G associated with θ with factor set α and P 1 of H asso-ciated with θ1 with factor set α1 such that α1ph1, h2q “ αph1, h2q for everyh1, h2 P H. Also, there is a map ν : CGpNq Ñ Fˆ such that Ppcq “ νpcqIθp1qand P 1pcq “ νpcqIθ1p1q for every c P CGpNq. Notice that if λ P IBrpZpNqq

lies under θ and θ1, then νZpNq affords λ. Also νpczq “ νpcqνpzq for everyc P CGpNq and z P ZpNq.

In order to construct projective representations P of G associated toθ and P 1 of H associated to θ1, we need some ingredients: appropriatetransversals T and T of the NCGpNq-cosets in N and the NCGpNq-cosets

in G, and a function ν : CGpNq Ñ Fˆ playing the role of ν for P and P 1.Let T Ď H be a complete set of representatives of cosets of NCGpNq

in G with 1 P T (we can choose such T because of G “ NH). Hence everyg P G can be written as tcn for some t P T, c P CGpNq and n P N . Noticethat tcn “ t1c1n1 if and only if there is some z P ZpNq such that t “ t1,c “ c1z, and n “ z´1n1. By elementary group theory, observe that T is alsoa complete set of representatives of cosets of MCGpNq in H.

Since HCGpNq and HCGpNq are equal as subgroups of AutpNq, for

every t P T we can choose t P H such that mt “ mt for every m P N . Alsowe can set 1 “ 1. By definition, we have that τptCGpNqq “ tCGpNq. Using

that τ is a group isomorphism, we have that T “ tt | t P Tu is a complete

set of representatives of right cosets of NCGpNq in G. Also since T is a

complete set of representatives of cosets of MCGpNq in H, we have that Tis a complete set of representatives of cosets of MCGpNq in H.

Let ν : CGpNq Ñ Fˆ be any function such that νpczq “ νpcqνpzq for ev-

ery c P CGpNq and z P ZpNq. (For instance, write CGpNq “Ťtj“1 xjZpNq,

and define νpxjzq “ νpzq.)We define functions

P : GÑ GLθp1qpF q and P 1 : H Ñ GLθ1p1qpF q

by

Pptncq “ PptqPpnqνpcq and P 1ptmcq “ P 1ptqP 1pmqνpcq,where t P T, n P N , c P CGpNq and m PM .

Notice that if tnc “ t1n1c1, then t “ t1, c “ c1z and n “ z´1n1 for somez P ZpNq, by a previous argument. Then

Ppn1c1q “ Ppn1qνpc1q “ Ppnqνpzqνpz´1cq “ Ppnqνpcq “ Ppncq,

using our defining property for ν. Hence P is well-defined. Similarly, oneproves that P 1 is a well-defined function. Notice that Ppnq “ Ppnq and

P 1pmq “ P 1pmq for every n P N and m PM .

We want to show that P and P 1 define projective representations of Gand H, associated with θ and θ1, respectively, with factor sets α and α1



coinciding on H ˆ H, and such that for x P CGpNq they are associated

with the same scalar. The latter part is obvious by the definition of Pand P 1. Let β be the factor set associated with P or with P 1. We recallthat βph, nq “ 1 “ βpn, hq for every h P H and n P N and, in particular,βph, h´1nhq “ 1.

Let x P G and let n P N . We prove that Ppxnq “ PpxqPpnq. Writex “ tmc, where m P N and c P CGpNq. Then xn “ tpmnqc and

Ppxnq “ Pptpmnqcq “ PptqPpmnqνpcq “ PptqPpmqνpcqPpnq “ PpxqPpnq .

Next, we prove that Ppnxq “ PpnqPpxq. By the comment in the previousparagraph about the factor set of P, we have that PpnqPptq “ PptqPpt´1ntq.Then

Ppnxq “ Ppntmcq “ Pptt´1ntmcq “ PptqPpt´1ntqPpmqνpcq“ PptqPpt´1ntqPpmqνpcq “ PpnqPptqPpmqνpcq

“ PpnqPpxq.

Next we show that P is a projective representation of G. Suppose thatt1, t2, t3 P T, c1, c2, c3 P CGpNq, and n1, n2, n3 P N are such that

pt1n1c1qpt2n2c2q “ t3n3c3.

Notice that

τpt1n1t2n2CGpNqq “ pt1n1pt2n2CGpNq “pt3n3CGpNq “ τpt3n3CGpNqq .

Thus t1n1t2n2 “ t3n3c, for some c P CGpNq. Then

Ppt1n1c1qPpt2n2c2q “ Ppt1n1qνpc1qPpt2n2qνpc2q

“ Ppt1n1t2n2qαpt1, t2qνpc1qνpc2q

“ Ppt3n3cqαpt1, t2qνpc1qνpc2q

“ Ppt3n3qPpcqαpt3, cq´1αpt1, t2qνpc1qνpc2q

“ Ppt3n3qνpc3qµpcqαpt3, cq´1αpt1, t2qνpc1qνpc2qνpc3q

´1.

This implies that P is a projective representation of G with factor set

αpt1n1c1, t2n2c2q “ µpcqαpt3, cq´1αpt1, t2qνpc1qνpc2qνpc3q

´1,

where c is any element of CGpNq satisfying the equation t1n1t2n2 “ t3n3c.The same argument, substituting elements in N by elements in M , showsthat P 1 is a projective representation of H with factor set

α1pt1m1c1, t2m2c2q “ µpcqαpt3, cq´1α1pt1, t2qνpc1qνpc2qνpcq

´1,

where c P CGpNq satisfies t1m1t2m2 “ t3m3c. It is now clear that both

factor sets coincide on H ˆ H. This finishes the proof.

Next, we discuss the Brauer characters of a central product of groupsand their relation with centrally isomorphic modular character triples.



Lemma 4.27. Let N Ÿ G and Ti with N ď Ti ď G be such that GN “

T1N ˆ ¨ ¨ ¨ ˆ TkN . Suppose that rTi, Tjs “ 1 for every i ‰ j. Givenθ P IBrpNq and ϕi P IBrpTi|θq, there is a unique χ “ ϕ1 ¨ . . . ¨ ϕk P IBrpG|θqsuch that χTi is a multiple of ϕi. Moreover, the map

IBrpT1|θq ˆ ¨ ¨ ¨ ˆ IBrpTk|θq Ñ IBrpG|θq

pϕ1, . . . , ϕkq ÞÑ ϕ1 ¨ . . . ¨ ϕk

is a natural bijection.

Proof. This is a natural adaptation of Lemma 5.1 of [IMN07] toBrauer characters.

In the situation described in Lemma 4.27, if ϕ1 P IBrpT1|θq,. . . ,ϕk PIBrpTk|θq, we refer to ϕ1 ¨ . . . ¨ ϕk P IBrpG|θq as the dot product of ϕ1, . . .,ϕk.

Lemma 4.28. Let N Ÿ H ď G. Suppose that pH,N, θq ąBr,c pK,M, θ1q.Let Z Ÿ G be an abelian group such that Z ď CGpNq and Z XN “ Z XM .Then

pHZ,NZ, θ ¨ νq ąBr,c pKZ,MZ, θ1 ¨ νq

for every ν P IBrHpZ|λq where λ P IBrpθZXM q.

Proof. See the proof of Proposition 3.9(b) of [NS14] together withTheorem 4.26.

The following method for constructing projective representations fromrepresentations given in [NS14] will be useful later in this work. Moreprecisely, it will be useful to control the values of certain projective repre-sentations in Section 4.3 (see the proof of Lemma 4.34).

We recall that if ε : G Ñ G is an epimorphism with kerpεq “ Z, then a

Z-section rep: GÑ G of ε is a map such that ε˝ rep “ idG and repp1q “ 1.

Theorem 4.29. Let pG,N, θq be a modular character triple. There exists

a finite group G, an epimorphism ε : G Ñ G with cyclic kernel Z ď ZpGq

and a Z-section rep: GÑ G satisfying:

(a) N “ N1 ˆ Z “ ε´1pNq, N1 – N via δ “ ε|N1 and N1 Ÿ G. The

action of G on N coincides with the action of G on N via ε.

(b) The character θ1 “ θδ´1P IBrpN1q extends to G. The Z-section

rep: G Ñ G satisfies reppnq P N1, reppngq “ reppnqreppgq andreppgnq “ reppgqreppnq for every n P N and g P G.

(c) εpCGpNqq “ CGpNq.

In particular, if D is a representation of G such that D|N1 affords θ1, thenthe map P defined for every g P G by

Ppgq “ Dpreppgqq

is a projective representation of G associated to θ.

Proof. See the proof of Theorem 4.1 of [NS14].


72 4.3. Fake Galois action on Brauer characters

4.3. Fake Galois action on Brauer characters

Perhaps, the most important application of Isaacs’ Bπ-theory (see Section1.4) relies on the fact that Bp1-characters constitute a canonical lift of Brauercharacters of p-solvable groups.

Theorem 4.30 (Isaacs). Let G be a p-solvable group. Then, restrictionto p-elements yields a bijection from Bp1pGq onto IBrpGq. In particular, theGalois group GalpQ|G|Qq acts on IBrpGq.

Proof. For the first part see Corollary 10.3 of [Isa84]. Then the latterstatement follows directly from Theorem 1.25.

However, in general, GalpQ|G|Qq does not act on IBrpGq. The groupG “ SL2p7q and the prime p “ 7 exemplify this, as pointed out to us byP. H. Tiep. Let σ P GalpQ|G|Q|G|21 q be the automorphism sending a 16-th

root of unity ξ to ξ3. By using GAP, we see that if ϕ P IrrpGq has degree 6,then ϕσ is not irreducible.

Let N be a group and let m be a positive integer. If θ is a class functionof N (or of N0), then θpmqpnq “ θpnmq is a class function of N (respectively

of N0). If p|N |,mq “ 1 and θ P IrrpNq, then θpmq “ θσ for a certain σ P

GalpQ|N |Qq and so θpmq P IrrpNq. It is no longer true that θpmq P IBrpNqif we consider θ P IBrpNq. (We refer to the above example for m “ 211.)

The aim of the final part of this works is to reduce Conjecture 5.1 to aproblem on quasi-simple groups. In order to do that, we will need that forevery quasi-simple group X, for every integer m with p|X|,mq “ 1 and forevery ϕ P IBrpXq, there exists some ϕ1 P IBrpXq that in some sense behaves

like ϕpmq. This is what we call a fake m-th Galois conjugate of ϕ. Let usstate this definition clearly. The results of this section are part of an originaljoint work with B. Spath [SV16].

Let V “ tξ P C | opξq “ n for some natural nu ď Cˆ. Recall that U isthe subgroup of p1-roots of unity of C. Hence U ď V. For a fixed positiveinteger m, let π be the set of primes dividing m. Define σm : V Ñ V by

σmpξq “ ξσm “ ξπξmπ1 ,

for every root of unity ξ P V, where ξπ and ξπ1 are respectively the π-partand the π1-part of ξ. Notice that σm is an automorphism of V and restrictionof σm to elements of U defines an automorphism of U which we denote againby σm. Let ωm : Fˆ Ñ Fˆ be the group homomorphism that σm inducesvia ˚ : U Ñ Fˆ. We denote by ζωm the image of ζ P Fˆ under ωm.

Moreover, for any positive integer n, by elementary Galois Theory, wehave that σm defines a Galois automorphism of Qn (which we denote againby σm P GalpQnQq) that sends every n-th root of unity ξ P Qn to ξσm .

Definition 4.31. Let pG,N,ϕq and pG,N,ϕ1q be modular charactertriples and let m be a positive integer coprime to |N |. We write

pG,N,ϕqpmq « pG,N,ϕ1q,



if there exist projective representations P and P 1 of G associated to ϕ andϕ1 such that:

(i) for every x, y P G the factor sets α and α1 of P and P 1 satisfy

αpx, yqωm “ α1px, yq,

and(ii) for every c P CGpNq the scalar matrices Ppcq and P 1pcq are associ-

ated with scalars ζ and ζωm .

In the above situation we may say that ϕ1 is a fake m-th Galois conjugateof ϕ with respect to N Ÿ G and that the triples pG,N,ϕq and pG,N,ϕ1q arefake m-th Galois conjugate.

Remark 4.32. Let pG,N,ϕq be a modular character triple, and let mbe a positive integer coprime to |N |.

(a) If ϕ is linear, then ϕσm “ ϕm P IBrpNq.

(b) In general ϕσm “ ϕpmq is an integer linear combination of IBrpNq.

(c) Let ϕ1 P IBrpNq and λ P IBrpϕ|ZpNqq. Then pN,N,ϕqpmq « pN,N,ϕ1q

if and only if IBrpϕ1|ZpNqq “ tλmu.

Proof. For part (a), notice that Qpϕq Ď Q|N |, then ϕσmpnq “ ϕpnqσm “

ϕpnqm for every n P N0, by the definition of σm. Of course, ϕm P IBrpNq,in this case. For part (b), let n P N0. Since ϕxny “ λ1 ` ¨ ¨ ¨ ` λt, whereλi P IBrpxnyq are linear, we have that

ϕpnqσm “ λ1pnqσm ` ¨ ¨ ¨ ` λtpnq

σm

“ λ1pnmq ` ¨ ¨ ¨ ` λtpn

mq

“ ϕpnmq “ ϕpmqpnq.

The class function ϕpmq is an integer linear combination of IBrpNq by Prob-lem 2.11 of [Nav98]. Part (c) follows from the definition of fake m-th Galoisconjugate modular character triples.

We shall give an alternative reformulation of Definition 4.31 in Lemma4.35. This new way of defining fake Galois conjugate modular charactertriples will make it easier to work with them later on. We first need to showthat for a modular character triple pG,N,ϕq there always exists a projectiverepresentation associated to ϕ with particular properties.

Lemma 4.33. Let N Ÿ G and χ P IBrpGq with χN P IBrpNq. ThenCGpNq

1 ď kerpχq.

Proof. Let D be a representation affording χ and c P CGpNq. ThenDpcq commutes with the irreducible representation DN . By Schur’s Lemma[Isa76, Lem. 1.5], this implies that Dpcq “ λpcqI is a scalar matrix. Themap λ : CGpNq Ñ Fˆ given by Dpcq “ λpcqI is a homomorphism. HenceCGpNqkerpλq is an abelian p1-group. This proves the statement.



Lemma 4.34. Let pG,N,ϕq be a modular character triple, m an integercoprime to |N | and pFˆqm1 the subgroup in Fˆ of elements of order coprimeto m. Then, there exists a projective representation P of G associated to ϕwith factor set α such that:

(i) αpg, g1q P pFˆqm1 for every g, g1 P G, and(ii) for every c P CGpNq, Ppcq is the scalar matrix associated to some

ξ P pFˆqm1.

Proof. Let ρ be the set of primes dividing m different from p, so thatp R ρ. (Then ρ “ π ´ tpu with the notation of this section.)

Suppose that ϕ extends to some χ P IBrpGq. We claim that there existsan extension ψ of ϕ to G such that every ρ-element of CGpNq lies in kerpψq.

By Lemma 4.33, we may assume that CGpNq1 “ 1. Hence CGpNq

is abelian and the Hall ρ-subgroup C of CGpNq satisfies C Ÿ G. ThenN X C “ 1 and NC – N ˆ C. We want to prove that there exists anextension of ϕ to G containing C in its kernel. It suffices to prove that thecharacter ϕ P IBrpNCCq given by ϕ (more precisely ϕpncCq “ ϕpnq forevery n P N and c P C) extends to GC.

Let q be any prime. If q P ρ and QN P SylqpGNq, then ϕ P IBrpNCCqextends to QCC because of pq, |NC : C|q “ 1 by Theorem 4.12. If q R ρ andQN P SylqpGNq we have that QXC “ 1. Since CŸ G, then χQ P IBrpQqdefines an irreducible Brauer character of QCC – Q which extends ϕ. ByTheorem 4.11, this implies that ϕ extends to GC, and the claim follows.

Now, by Theorem 4.29, there exists a central extension ε : G Ñ G of Gwith finite cyclic kernel Z and a Z-section rep: GÑ G of ε such that:

(a) N “ N1 ˆ Z “ ε´1pNq, the groups N1 and N are isomorphic via

δ “ ε|N1 and N1 Ÿ G. Moreover, the action of G on N coincideswith the action of G on N via ε.

(b) ϕ1 “ ϕδ´1P IBrpN1q extends to G. The Z-section rep: G Ñ G

satisfies reppnq P N1, reppngq “ reppnqreppgq and reppgqreppnq “reppgnq for every n P N and g P G.

(c) εpCGpNqq “ CGpNq.

According to (b) the character ϕ1 extends to G. By the first part of the

proof applied to G, there is an extension χ1 P IBrpGq such that every ρ-

element of CGpNq lies in kerpχ1q. Let D be a representation affording χ1

and let P : GÑ GLϕp1qpF q be defined by

Ppgq “ Dpreppgqq for every g P G.

For every g, g1 P G we obtain

PpgqPpg1q “ Dpzg,g1qPpgg1q,

where zg,g1 P Z ď ZpGq is given by reppgqreppg1q “ zg,g1reppgg1q. SinceDpzg,g1q is a scalar matrix, P is a projective representation of G associated



to ϕ with factor set α : G ˆ G Ñ Fˆ defined by αpg, g1qI “ Dpzg,g1q forevery g, g1 P G. Since Dpcq “ Iϕ1p1q for every ρ-element c P CGpNq, it isstraightforward to check that P satisfies the required properties.

As a consequence of Lemma 4.34, we can reformulate Definition 4.31 inthe following (easier to handle) way.

Lemma 4.35. Let pG,N,ϕq and pG,N,ϕ1q be modular character triples,and let m be coprime to |N |. Then the following are equivalent:

(a) pG,N,ϕqpmq « pG,N,ϕ1q,(b) there exist projective representations P and P 1 of G associated to ϕ

and ϕ1 satisfying the properties (i) and (ii) in 4.34 and such that(b.1) the factor sets α and α1 of P and P 1 satisfy

αpg, g1qm “ α1pg, g1q for every g, g1 P G,

and(b.2) for every c P CGpNq, the scalar matrices Ppcq and P 1pcq are

associated with scalars ζ and ζm respectively.

Proof. We first prove that (a) implies (b). Since pG,N,ϕqpmq « pG,N,ϕ1qthere exist projective representations Q and Q1 of G associated to ϕ and ϕ1

having the properties listed in Definition 4.31. Let P be a projective repre-sentation of G associated to ϕ with the properties described in Lemma 4.34.By Remark 4.15, there exists a map µ : G Ñ Fˆ such that P is similar toµQ. Also µ is constant on N -cosets in G and µp1q “ 1. Let µ1 : GÑ Fˆ begiven by µ1pgq “ µpgqωm for every g P G. Since µ1 is constant on N -cosetsin G and µ1p1q “ 1, then it is straightforward to check that P 1 “ µ1Q1 is aprojective representation of G associated to ϕ1.

In order to verify that P and P 1 satisfy the condition in (b.1) let g, g1 P G.According to Definition 4.31, the factor sets β and β1 of Q and Q1 satisfyβpg, g1qωm “ β1pg, g1q. By Remark 4.15, the factor sets α and α1 of P andP 1 satisfy

αpg, g1q “µpgqµpg1q

µpgg1qβpg, g1q and α1pg, g1q “

µ1pgqµ1pg1q

µ1pgg1qβ1pg, g1q.

This implies that αpg, g1qωm “ α1pg, g1q. By the choice of P, we have thatαpg, g1q is a root of unity in Fˆ of order coprime to m, hence α1pg, g1q “αpg, g1qωm “ αpg, g1qm also is. This proves that P and P 1 satisfy the condi-tion (b.1).

In order to verify that P and P 1 satisfy the condition in (b.2) let c PCGpNq and ζ P Fˆ be the scalar associated to Qpcq. According to Definition4.31, ζωm is the scalar associated to Q1pcq . By definition Ppcq and P 1pcq arescalar matrices associated with ζµpcq and pζµpcqqωm respectively. By thechoice of P, we have that ζµpcq is a root of unity of Fˆ of order coprime tom. Hence pζµpcqqωm “ pζµpcqqm P Fˆ has also order coprime to m. Thisproves that P and P 1 satisfy the condition (b.2).



Moreover we see that P 1 is a projective representation having the prop-erties mentioned in 4.34.

To prove the converse, just notice that the projective representations Pand P 1 in (b) have the properties described in Lemma 4.34 and recall thatωm acts on roots of unity of Fˆ of order coprime to m by raising them totheir m-th power. It is then immediate that P and P 1 give pG,N, θqpmq «pG,N, θ1q.

The following is an easy consequence of Lemma 4.35.

Corollary 4.36. Let pG,N,ϕq be a modular character triple. Let m bea positive integer coprime to |N | and let σm be defined as in the beginning

of this section. If ϕ is linear, then pG,N,ϕqpmq « pG,N,ϕσmq.

Proof. By Remark 4.32, we know that ϕσm “ ϕm. Let P be a pro-jective representation of G associated to ϕ as in Lemma 4.34. Then, itis straightforward to check that Pm is a projective representation of Gassociated to ϕm (as in Lemma 4.34). By Lemma 4.35, we have that

pG,N,ϕqpmq « pG,N,ϕσmq.

We have defined the notion of fake m-th Galois conjugate charactertriples. We conclude this section by introducing fake m-th Galois actionsand verifying their existence on p-solvable groups.

Definition 4.37. Let NŸ G. Let S ď IBrpNq be a G-invariant subset.Let m be an integer coprime to |N |. We say that there exists a fake m-thGalois action on S with respect to G if there exists a G-equivariantbijection

fm : S Ñ S

such that

pGϕ, N, ϕqpmq « pGϕ, N, fmpϕqq for every ϕ P S .

Let N be a p-solvable group. Then the Galois group GalpQ|N |Qq actson IBrpNq, by Theorem 4.30. In this case, we show that there exists a fakem-th Galois action on IBrpNq for any integer m coprime to |N | and withrespect to any G with N Ÿ G. We first need to control the values of certainprojective (ordinary) representations.

We recall that the theory of projective representations associated to acharacter triple over C is very similar to the theory of projective represen-tations associated to character triples over F . In particular, Theorem 8.12and Lemma 8.27 of [Nav98] work for C instead of F .

Recall we have chosen an ideal M ă R with pR ĎM and F “ RM.

Lemma 4.38. Let N Ÿ G and θ P IrrpNq. Assume that θ is G-invariantand θ0 P IBrpNq. Write L “ Q|G| and SL “ trs | r P RXL, s P RXLzMu.There exists a projective representation of G associated to θ with matrixentries in SL.



Proof. We notice that by Brauer’s Theorem [Isa76, Thm. 10.3], Lis a splitting field for G. By Problem 2.12 of [Nav98], let Y be an SL-representation of N affording θ. We check that there exists a projectiverepresentation of G associated to θ extending Y with entries in SL. Forevery g P GN , the representation Y extends to a representation Yg ofxN, gy as in Theorem 8.12 of [Nav98], where g P G with gN “ g. DefineDpgq “ Ygpgq for every g P G. Then D is a projective representation of Gextending Y, by Lemma 8.27 of [Nav98]. Hence, it suffices to control thematrix entries of the representations Yg. In other words, we only need tocheck the case where θ extends to G.

Let χ P IrrpGq be an extension of θ. Let X be a representation affordingχ with entries in SL (again such X does exist by Problem 2.12 of [Nav98]).We have that XN affords θ. Hence, there is some T P GLnpLq such thatXN “ T´1YT . Write T “ ptijq, where tij P L. By Lemma 2.5 of [Nav98],since all tij are algebraic over Q, there exists β P L such that all βtij P R butnot all βtij P M. Since XN “ pβT q

´1YpβT q, we may assume that tij P Rand T ˚ ‰ 0 (replacing T by βT ). By assumption, the F -representationspXN q

˚ and Y˚ are irreducible. Moreover T ˚pXN q˚ “ Y˚T ˚. By Schur’s

Lemma [Isa76, Lem. 1.5], this implies T ˚ P GLnpF q. In particular, wehave that detpT ˚q “ detpT q˚ ‰ 0 and so detpT q R M. Thus T P GLnpSLqand the representation TXT´1 with entries in SL extends Y.

Theorem 4.39. Let N be a p-solvable group and let m be a positiveinteger coprime to |N |. If N Ÿ G, then there exists a fake m-th Galoisaction on IBrpNq with respect to G.

Proof. By Theorem 4.30, we have that Bp1pNq provides a canonicallift of IBrpNq. Moreover, AutpNq and GalpQ|N |Qq act on the set Bp1pNqby Theorem 1.25. Thus, the bijection Bp1pNq Ñ IBrpNq given by θ ÞÑθ0 commutes with the action of AutpNq. Consider σm as defined at thebeginning of this section, before Definition 4.31. Let ϕ P IBrpNq and θ PBp1pNq with ϕ “ θ0. Since θσm P Bp1pNq, we have that ϕσm “ pθσmq0 PIBrpNq. Hence the map fm : IBrpNq Ñ IBrpNq defined by ϕ ÞÑ ϕσm is anAutpNq-equivariant bijection.

Let ϕ P IBrpNq. We want to prove that pGϕ, N, ϕqpmq « pGϕ, N, ϕ

σmq.We may assume that G “ Gϕ. Let θ P Bp1pNq be the canonical lift of ϕ.Then θ is G-invariant and θσm is the canonical lift of ϕσm . Write L “ Q|G|.By Lemma 4.38, there exists a projective representation D of G associatedto θ with matrix entries in SL “ trs | r P R X L, s P R X LzMu. Inparticular, the map Dσm is a well-defined projective representation of Gassociated to θσm with matrix entries in SL. It is straightforward to checkthat the F -projective representations P “ D˚ and P 1 “ pDσmq˚ associatedto ϕ and ϕσm satisfy the required properties of Definition 4.31.


CHAPTER 5

Coprime action and Brauer characters

5.1. Introduction

Let A and G be finite groups. Assume that A acts coprimely on G (recallthat this means that A acts by automorphism on G and p|A|, |G|q “ 1).Then there exists a canonical bijection

πpG,Aq : IrrApGq Ñ IrrpCGpAqq,

between the set IrrApGq of irreducible characters fixed under the action of Aand the irreducible characters of the group CGpAq of fixed points under theaction of A. This bijection is known as the Glauberman-Isaacs correspon-dence. (We refer the reader to the discussion preceding Theorem 1.17 andto Theorem 1.17 itself.) The bare fact that these two sets have the samecardinality is highly non-trivial and has already important consequences.For instance, it proves that the actions of A on the irreducible characters ofG and on the conjugacy classes of G are permutation isomorphic.

Let us fix a prime p. Recall that g P G is p-regular if p does not divideopgq. In this chapter we consider the same question on p-Brauer charactersand p-regular conjugacy classes.

Conjecture 5.1. Let A and G be finite groups. Let p be a prime.Suppose that A acts coprimely on G. Then the actions of A on the irreducibleBrauer characters of G and on the conjugacy classes of p-regular elementsof G are permutation isomorphic.

Conjecture 5.1 is an open problem posed by G. Navarro in [Nav94].By Corollary 13.10 and Lemma 13.23 of [Isa76], it is easy to prove thatConjecture 5.1 is equivalent to the following.

Conjecture 5.2. Let A and G be finite groups. Let p be a prime.Suppose that A acts coprimely on G. Then the number of A-invariant irre-ducible Brauer characters of G is equal to the number of irreducible Brauercharacters of CGpAq.

There is some evidence of the validity of Conjecture 5.2. First, it holdswhenever G is p-solvable by work of K. Uno [Uno83] (fundamentally basedon the fact that there exist canonical lifts of the irreducible Brauer char-acters). The fact that Conjecture 5.2 holds whenever A is a cyclic group,and hence also Conjecture 5.1, is a well-known consequence of the so-calledBrauer’s argument on the character table (see Lemma 1.16; the argument

79


is the same in both the ordinary and the modular cases). In particular,this implies that Conjecture 5.1 holds whenever G is a quasi-simple group,using the classification of the finite simple groups (see Theorem 5.17). Ithas also been proven in [NST16] that A fixes a unique irreducible Brauercharacter of G if and only if CGpAq is a p-group, proving Conjecture 5.2 inthis case. However, if G is not p-solvable (and therefore A is solvable) noother progress has been made.

Our aim in this chapter is to reduce Conjecture 5.2 to a problem on finitesimple groups (in the same spirit as for the McKay conjecture [IMN07]).We can prove the following, which is the main result of this chapter.

Theorem G. Let G and A be finite groups. Suppose that A acts co-primely on G. Suppose that all finite non-abelian simple groups (of orderdivisible by p) involved in G satisfy the inductive Brauer Glauberman con-dition. Then the number of irreducible p-Brauer characters of G fixed by Ais the number of irreducible p-Brauer characters of CGpAq.

Consequently, the actions of A on the irreducible p-Brauer charactersand on the conjugacy classes of p-regular elements of G are permutationisomorphic.

Hence, we prove that if every non-abelian simple group satisfies the in-ductive Brauer-Glauberman condition, then Conjectures 5.1 and 5.2 hold.We do not define this inductive condition right now, but we feel it is worthmentioning that there is a surprising difference between this inductive con-dition and other inductive conditions coming from global/local conjectures(for instance the inductive McKay condition in [IMN07]). This differenceis derived from the fact that the Galois group does not act on irreducibleBrauer characters together with the fact that Galois action plays an im-portant role in the description of the Glauberman correspondent in a keysituation. We will require the existence of fake Galois actions in our induc-tive condition. See Definition 5.24 for further details.

The content of this chapter is arduous. We think that it is fair to saythat our reduction of Conjecture 5.2 is at least as hard as the reduction ofthe McKay conjecture. Although both reductions bear similarities, thereare also differences, as we said. It does not seem possible to conduct bothreductions at the same time. Our reduction here is used in the forthcomingpaper [NST16] in which more evidence for the truth of Conjectures 5.1 and5.2 is given.

This chapter is structured in the following way: In Section 5.2 we re-call some well-known results on character counts above Glauberman-Isaacscorrespondents. In Section 5.3 we study a particular case of the Glauber-man correspondence. This particular example motivates the definition offake Galois conjugate modular character triples in Section 4.3. We also ex-plain how to construct centrally isomorphic modular character triples fromfake Galois conjugate ones. In Section 5.4 we study coprime actions on


5. Coprime action and Brauer characters 81

direct products of quasi-simple groups. In Section 5.5 we define the in-ductive Brauer-Glauberman condition on finite simple groups. Then, we as-sume that a finite non-abelian simple group S satisfies the inductive Brauer-Glauberman condition and we study consequences for the character theoryof central extensions of direct products S ˆ ¨ ¨ ¨ ˆ S of S. This section isof a highly technical nature and the results contained in it are key in theinductive process used to prove Theorem G. In Section 5.6, making useof everything proved in preceding sections, we can prove Theorem G. Weconclude by studying some natural questions related to Conjecture 5.2 inSection 5.7.

All the results of this chapter are part of an original work of the authortogether with B. Spath [SV16].

5.2. Review on character counts above Glauberman-Isaacscorrespondents

The main goal of this section is to count Brauer characters lying abovecharacters of normal p1-subgroups and their Glauberman correspondents.

We shall use the following.

Lemma 5.3. Assume that a group A acts on a group G. Let K Ÿ Gbe A-invariant. Assume that p|G : K|, |A|q “ 1 and CGKpAq “ GK. Ifη P IBrApKq, then every χ P IBrpG|ηq is A-invariant.

Proof. The proof follows from the same arguments as the in proof ofLemma 2.5 of [Wol78a].

Suppose that K Ÿ G and η is an irreducible G-invariant character ofK. If η is an ordinary character, then |IrrpG|ηq| can be determined bya purely group theoretical method due to P. X. Gallagher. There is ananalogous result for Brauer characters. If η is a Brauer character we say thatKg P pGKq0 (or g) is η-good if every extension ϕ P IBrpxK, gyq of η is U -invariant where UK “ CGKpKgq. Notice that η always extends to xK, gysince xK, gyK is cyclic (by Theorem 4.10). Also, U acts on IBrpxK, gy|ηqsince xK, gy U . It is clear that if Kg P pGKq0 is η-good, then everyG-conjugate of Kg also is, so we can talk about p-regular η-good classes ofGK.

Theorem 5.4. Suppose that KŸ G and that η P IBrpKq is G-invariant.Then, |IBrpG|ηq| is equal to the number of p-regular η-good classes of GK.


The following is basically a particular case of Theorem 2.12 of [Wol79].

Theorem 5.5. Let A act coprimely on G. Suppose that K Ÿ G is A-invariant and G “ KC, where C “ CGpAq. Then η P IrrApKq extendsto Gη iff η1 P IrrpK X Cq extends to Cη1, where η1 P IrrpCKpAqq is theGlauberman-Isaacs correspondent of η.


82 5.2. Review on character counts

Proof. Notice that GηXC “ Cη1 by Lemma 2.5(b) of [Wol79]. Hence,we may assume that η and η1 are C-invariant.

Suppose that η extends to some χ P IrrpGq. By [Wol78a, Lem. 2.5] wesee that χ P IrrApGq. Since rG,As ď K, by [Wol79, Thm. 2.12]

η1 “ πpK,Aqpηq “ πpK,AqpχKq “ pπpG,AqpχqqKXC .

Hence πpG,Aqpχq is an extension of η1. Analogously we see that η extends to

G, if η1 extends to C.

We need similar results for Brauer characters. We state the followingeasy observation as a lemma for the reader’s convenience.

Lemma 5.6. Let K G, θ P IrrpKq be G-invariant and suppose that Kis a p1-group. Then θ extends to an ordinary character of G if and only if θextends to a Brauer character of G.

Proof. Let χ P IrrpGq be an extension of θ to G. Then χ0 is a Brauercharacter of G extending θ since K Ď G0. In particular, χ0 P IBrpGq.Suppose that ϕ P IBrpGq extends θ. Let QN be a Sylow q-subgroup ofGN for some prime q. If q ‰ p, then ϕQ is an ordinary character extendingθ. If q “ p, then θ extends to Q by Theorem 1.13. Hence θ extends to G byTheorem 1.15.

Theorem 5.7. Suppose that A acts coprimely on G. Let K Ÿ G be anA-invariant p1-group. Suppose that G “ KC, where C “ CGpAq. Let η PIrrApKq be G-invariant and write η1 P IrrpKXCq to denote its Glauberman-Isaacs correspondent. Then

|IBrpG|ηq| “ |IBrpC|η1q|.

Proof. By Theorem 5.4 it suffices to show that for every c P C, theelement cK is η-good if and only if the element cK X C is η1-good. ByTheorem 4.7 of [IN96], it suffices to show that for every U with K ď U ď Gand abelian UK, η extends to U as a Brauer character if and only if η1

extends to UXC as a Brauer character. We apply Theorem 5.5 and Lemma5.6 in U . Then we are done.

Corollary 5.8. Suppose that A acts coprimely on G. Let K Ÿ G bean A-invariant p1-group. Suppose that G “ KC, where C “ CGpAq. LetN Ÿ G be contained in K X C and let θ P IrrpNq. Then

|IBrApG|θq| “ |IBrpC|θq|.

Proof. Let B be a set of representatives of the C-orbits of IrrApK|θqand B1 “ tπpK,Aqpηq | η P Bu. Then B1 Ď IrrpKXC|θq by [Wol79, Lem. 2.4]

and B1 is a set of representatives of the C-orbits of IrrpKXC|θq. By Corollary5.2 of [Wol78a], we deduce that the bijection πpK,Aq is C-equivariant. Hence

B1 is a set of representatives of the C-orbits of IrrpK X C|θq.



Every element of IBrApG|θq lies over a unique element of B and alsoevery element of IBrpC|θq lies over a unique element of B1. Thus

|IBrApG|θq| “ÿ

ηPB|IBrApG|ηq| and |IBrpC|θq| “

ÿ

ηPB|IBrpC|πpK,Aqpηqq|.

By Lemma 5.3, for every η P B we have that |IBrApG|ηq| “ |IBrpG|ηq|.Hence, it suffices to show that |IBrpG|ηq| “ |IBrpC|πpK,Aqpηqq| for everyη P B. By the Clifford correspondence on Brauer characters, we may assumethat η P B is G-invariant. Now, the result follows from Theorem 5.7.

5.3. More on fake Galois conjugates

We begin this section with the description of the Glauberman correspon-dence in a very specific situation. This example shows that the Glaubermancorrespondence is related to a certain Galois action on ordinary characters.

Let G be a group and let A ď Sm. Recall Gm denotes the external

direct product of m copies of G. Write rG “ Gm. Then A acts on rG by

pg1, . . . , gmqa´1

“ pgap1q, . . . , gapmqq for gi P G and a P A. If A is transitive,then

CrGpAq “ tpg, . . . , gq P rG | g P Gu.

The group CrGpAq is isomorphic to G. An irreducible A-invariant character

χ of rG has the form

χ “ θ ˆ ¨ ¨ ¨ ˆ θ

for some θ P IrrpGq. If p|A|, |G|q “ 1, then the Glauberman correspondent

of an A-invariant character χ “ θˆ ¨ ¨ ¨ ˆ θ of rG is some Galois conjugate ofθ viewed as a character of C

rGpAq. (See Proposition 5.10 below.)

Notation 5.9. Let m be a positive integer. Recall the definition ofσm from Section 4.3: let π be the set of primes dividing m, then for everypositive integer n we define σm P GalpQnQq to be the automorphism fixingπ-roots of unity and raising to the m-th power π1-roots of unity. For a group

G, we denote here by rG the external direct product Gm of m copies of G.

Let ∆m : G Ñ rG be the injective morphism defined by g ÞÑ pg, . . . , gq forevery g P G. Then ∆m defines natural bijections IrrpGq Ñ Irrp∆mGq andIBrpGq Ñ IBrp∆mGq, where ∆mG “ ∆mpGq. We also write ∆mθ “ ∆mpθq.If from the context m is clear, we omit the superscript m and write ∆, ∆Gand ∆θ instead of ∆m, ∆mG and ∆mθ.

Proposition 5.10. Let m be a positive integer. Assume that a solvablesubgroup A ď Sm is transitive. Let G be a finite group with p|A|, |G|q “

1. Let rG be the direct product of m copies of G. Let θ P IrrpGq and let

χ “ θ ˆ ¨ ¨ ¨ ˆ θ P IrrAp rGq. Then, the Glauberman correspondent of χ isthe character χ1 “ ∆θσm, where θσm is the image of θ under the Galoisautomorphism σm.


84 5.3. More on fake Galois conjugates

Proof. This is essentially the content of Exercise 13.11 of [Isa76]. Todo the case where A is cyclic of prime order, use Exercise 4.7 of [Isa76].The general case, follows by induction on |A|.

It is clear that the Clifford theory over two ordinary irreducible Galoisconjugate characters is related.

Proposition 5.11. Let N Ÿ G and θ P IrrpNq. Let m be a positiveinteger coprime to |N |. Let θ1 “ θσm. Then:

(a) Gθ “ Gθ1.(b) Assume G “ Gθ. There exist P and P 1 projective representations

of G associated to θ and θ1 such that:(b.1) the factor sets α and α1 of P and P 1 satisfy

αpg, g1qσm “ α1pg, g1q

for every g, g1 P G, and(b.2) for every c P CGpNq the scalar matrices Ppcq and P 1pcq are

associated with ξ and ξσm for some root of unity ξ P Q|G|.

Proof. By Theorem 4.29 and [Isa76, Thm. 10.3], there exists a pro-jective representation P of G associated to θ whose entries are in Qk forsome k ě 1. Choose P 1 “ Pσm . The result follows from straightforwardcalculations.

It is worth comparing Proposition 5.11 with Definition 4.31. If pG,N,ϕqand pG,N,ϕ1q are fake m-th Galois conjugate modular character triples,then the Clifford theory over ϕ and ϕ1 is related in the same way as theClifford theory over two σm-conjugate ordinary character triples.

We recall below the definition of fake Galois conjugate characters in thealternative version provided by Lemma 4.35.

Definition 5.12. Let pG,N,ϕq and pG,N,ϕ1q be modular charactertriples, and let m be a positive integer coprime to |N |. Then we say that ϕand ϕ1 are fake m-th Galois conjugate with respect to N Ÿ G, and we writepG,N,ϕqpmq « pG,N,ϕ1q, if there exist projective representations P and P 1of G associated to ϕ and ϕ1 with factor sets α and α1 such that

(i) αpg, g1q, α1pg, g1q P Fˆ have order coprime to m and

αpg, g1qm “ α1pg, g1q

for every g, g1 P G, and(ii) for every c P CGpNq, the scalar matrices Ppcq and P 1pcq are associ-

ated with elements ζ, ζ 1 P Fˆ of order coprime to m and ζm “ ζ 1.

The notion of fake Galois conjugate (modular) character triples is im-portant in our later application, since it allows us to construct centrallyisomorphic character triples from Galois conjugate character triples (seeTheorem 5.14 below).



Notation 5.13. Let m be a positive integer. For groups H ď G wecontinue writing Hm to denote the external direct product of m copies ofH. Recall that ∆H ď Gm. For groups K,H ď G with K ď NGpHq, we

denote by q∆HK the group Hm∆K ď Gm.

Recall the description of the Brauer characters of a central product ofgroups in Lemma 4.27.

Theorem 5.14. Let pG,N,ϕq and pG,N,ϕ1q be modular character triples.

Let Z “ ZpNq. Write rZ “ Zm, rN “ Nm, qG “ q∆ZG and qN “ q∆ZN . Let

ν P IBrpϕZq, rϕ “ ϕ ˆ ¨ ¨ ¨ ˆ ϕ P IBrp rNq, rν “ ν ˆ ¨ ¨ ¨ ˆ ν P IBrp rZq and

qϕ “ ∆ϕ1 ¨ rν P IBrp qNq. Then the following are equivalent:

(a) pG,N,ϕqpmq « pG,N,ϕ1q,

(b) p rNp qG¸Smq, rN, rϕq ąBr,c p qG¸Sm, qN, qϕq.

Proof. Notice that

rNp qG¸Smq “ p rN qGq ¸Sm.

Also rNXp qG¸Smq “ qN is the central product of rZ and ∆N . Hence, the char-

acters in IBrp qNq are dot products of characters in IBrp rZq and characters inIBrp∆Nq that lie over the same λ P IBrp∆Zq (see Theorem 4.27). Note that

qϕ “ ∆ϕ1 ¨ν is well-defined and lies in IBrp qNq since IBrprν∆Zq “ IBrpp∆ϕ1q∆ZqNow, we prove that (a) implies (b). Let Q and Q1 be projective repre-

sentations of G giving

pG,N,ϕqpmq « pG,N,ϕ1q

as in Definition 5.12.We construct projective representations P0 and P 10 of rN qG and qG associ-

ated to rϕ and qϕ respectively. Note that rN qG “ q∆NG. It is straightforward

to show that the map P0 : rN qGÑ GLϕp1qmpF q given by

P0ppn1, . . . , nmq∆gq “ Qpn1gq b ¨ ¨ ¨ bQpnmgq

for every pn1, . . . , nmq P rN and g P G, defines a projective representationassociated to rϕ. The factor set α0 of P0 satisfies

α0prn∆g, rn1∆g1q “ βpg, g1qm for every rn, rn1 P rN and g, g1 P G,

where β denotes the factor set of Q. Let τ be an F -representation of rZ

affording rν. The map P 10 : qGÑ GLqϕp1qpF q given by

P 10prz∆gq “ τprzqQ1pgq for every rz P rZ and g P G

is a projective representation associated to qϕ. The factor set α10 of P 10 satisfies

α10prz∆g, rz1∆g1q “ β1pg, g1q “ βpg, g1qm “ α0prz∆g, rz

1∆g1q

for every g, g1 P G and z, z1 P rZ, where β1 denotes the factor set of Q1.(Recall that β1pg, g1q “ βpg, g1qm since Q and Q1 satisfy Definition 5.12(b.1)).


86 5.3. More on fake Galois conjugates

In the next step we extend P0 to a projective representations P of

p rN qGq ¸ Sm. Note that Sm has a natural action on the tensor spaceÂ

Fϕp1q by permuting the tensors. This induces a representation R : Sm Ñ

GLϕp1qmpF q. The map P : q∆NG¸Sm Ñ GLϕp1qmpF q given by

Ppxσq “ P0pxqRpσq for every x P rN qG and σ P Sm

is a projective representation of rN qG¸Sm “ rNp qG¸Smq. Note that PN oSmis a representation, as defined in [Hup98, Thm. 25.6]. It is easy to check,using the definition of R, that the factor set α of P satisfies

αprn∆gσ, rn1∆g1σ1q “ α0prn∆g, rn1∆g1q

for every g, g1 P G, rn, rn1 P rN and σ, σ1 P Sm.

In the next step we extend P 10 to a projective representation P 1 of qG¸Sm.Note that r∆G,Sms “ 1. Hence rν and τ are Sm-invariant and the map

P 1 : qG ¸ Sm Ñ GLϕ1p1qpF q defined by P 1pgrzσq “ P 10pgqτprzq for every g PqG, rz P rZ and σ P Sm is a projective representation whose factor set α1

satisfiesα1pgσ, g1σ1q “ α10pg, g

1q

for every g, g1 P qG and σ, σ1 P Sm. In particular, we see that the projectiverepresentations P and P 1 that we have constructed satisfy the propertydescribed in Definition 4.19(ii.1).

In the last step we compare Ppxq and P 1pxq for x P Cp rNGq¸Sm

p rNq. We

have thatCp rNGq¸Sm

p rNq “ q∆ZCGpNq ď qG¸Sm.

Then x “ rz∆c for some rz P rZ and c P CGpNq. Recall that by Definition5.12(b.2), Qpcq and Q1pcq are scalar matrices associated with some ζ and ζm.Thus Pprz∆cq and P 1prz∆cq are scalar matrices associated with τprzqζm, bythe definition of P and P 1. This implies that P and P 1 satisfy the propertyin Definition 4.19(ii.2).

This proves (a) implies (b), since we have already seen that the grouptheory conditions of Definition 4.19 are satisfied.

We only sketch the steps to prove that (b) implies (a). We start bychoosing a projective representation Q of G associated to ϕ as in Lemma

4.34. Then one can construct a projective representation P of p rN qGq ¸Sm

associated to rϕ as in the first part of the proof. Let P 1 be the projective

representation of qG ¸ Sm associated to qϕ given by Lemma 4.21(a). ThenP 1|∆G defines via the natural isomorphism ∆G Ñ G a projective represen-tation Q1 of G associated to ϕ1, because P 1|∆N affords ∆ϕ1. It is easy to

check that Q and Q1 give pG,N,ϕqpmq « pG,N,ϕ1q using Lemma 4.35.

Corollary 5.15. Assume the notation and the situation of Theorem

5.14. Let H Ÿ G with H ď CGpNq and write qG1 “ q∆ZHG. Then

p rNp qG1 ¸Smq, rN, rϕq ąBr,c p qG1 ¸Sm, qN, qϕq.



Proof. Note that rN qG1 “ q∆NHG and qG1 “ q∆HZG are well-defined.The group q∆NHG¸Sm induces on rN the same automorphisms as q∆NG¸Sm. Hence, the statement follows from Theorem 5.14 after applying Theo-rem 4.26.

Also as a consequence of Theorem 5.14, we obtain the analogue of Theo-rem 4.26 for fake Galois conjugate modular character triples: pG,N,ϕqpmq «pG,N,ϕ1q is a property that only depends on the characters ϕ and ϕ1 as wellas the automorphisms induced by G on N .

Corollary 5.16. Let pG,N,ϕq and pG,N,ϕ1q be modular charactertriples. Suppose that for a positive integer m coprime to |N |,

pG,N,ϕqpmq « pG,N,ϕ1q.

Let G1 be a group such that N Ÿ G1 and G1CG1pNq is equal to GCGpNqas a subgroup of AutpNq. Then

pG1, N, ϕqpmq « pG1, N, ϕ

1q.

Proof. This follows from combining Theorem 5.14 and Theorem 4.26.

5.4. Coprime action on simple groups and their direct products

In this section we study the situation where a group A acts coprimely onthe direct product of isomorphic non-abelian simple groups (as well as onthe direct product of their universal covering groups). If a group A acts ona set Λ, then we write AΛ0 to denote the stabilizer of Λ0 Ď Λ in A, so that

AΛ0 “ ta P A | Λa0 “ Λ0u.

We begin by studying coprime actions on finite simple non-abelian groups.Let S be a non-abelian simple group and letX be its universal covering group(unique up to isomorphism). We identify AutpSq and AutpXq as in [Asc00,Ex. 6, Chapt. 11].

Theorem 5.17. Let S be a simple non-abelian group. Let B act on Sfaithfully with p|S|, |B|q “ 1. Then B is cyclic, NAutpSqpBq “ CAutpSqpBqand

ZpCXpBqq “ CXpBq X ZpXq,(5.1)

where X is the universal covering group of S and B acts on X via thecanonical identification AutpSq “ AutpXq.

Proof. We identify B with the corresponding subgroup of AutpSq. Wemay assume B ‰ 1, otherwise the result is trivial. According to the classi-fication of finite simple groups the group S has to be a simple group of Lietype and B is AutpSq-conjugate to some group of field automorphisms of S,


88 5.4. Coprime action on simple groups

see for example Section 2 of [MNS15]. In particular, B is cyclic. The struc-ture of AutpSq is described in Theorem 2.5.12 of [GLS03]. Straightforwardcomputations with AutpSq prove that NAutpSqpBq “ CAutpSqpBq.

Let X be the universal covering group of S. In order to prove thatZpCXpBqq “ CXpBq X ZpXq we may assume that B ‰ 1. By the firstparagraph of this proof S is a simple group of Lie type. Arguing as in thebeginning of Section 2 of [MNS15] we see that the Schur multiplier of S isgeneric or S “ 2B2p8q and B is a cyclic group of order 3.

Let us consider first the case where S “ 2B2p8q and that B is a cyclicgroup of order 3. Then X “ 22.2B2p8q and CXpBq “

2B2p2q. The group2B2p2q is a Frobenius group of order 5 ¨ 4 and has trivial centre. It followsthat ZpXq XCXpBq is also trivial.

Hence we can assume that X “ XF , for some simply-connected simplealgebraic group X and some Steinberg endomorphism F : X Ñ X. We canassume that B is generated by some automorphism that is induced by someSteinberg endomorphism F0 : X Ñ X. Without loss of generality we canassume that some power of F0 coincides with F . From Theorem 24.15 of[MT11] we deduce that

ZpCXpBqq “ ZpXF0q “ ZpXqF0pZpXqF qF0

“ pZpXF qqF0 “ ZpXq XCXpBq.

This is exactly Equation 5.1

Notation 5.18. Let S be a non-abelian finite simple group. Let X bethe universal covering group of S. Let r be a positive integer. Recall fromprevious sections that Xr denotes the external direct product of r copies of

X. We write rX “ Xr. Note that rX is also the internal direct product ofX1, . . . , Xr, where the group Xi is defined by

Xi “ 1ˆ ¨ ¨ ¨ ˆX ˆ ¨ ¨ ¨ ˆ 1

with an X at the i-th position, for each i P t1, . . . , ru.

Recall we identify AutpSq and AutpXq. Also AutpXq oSr acts on rX via

px1, . . . , xrqpα1,...,αrqσ “ ppxσ´1p1qq

α1 , . . . , pxσ´1prqqαrq,

for every xi P X, αi P AutpXq and σ P Sr. It is easy to show that this

defines an isomorphism between AutpXq o Sr and Autp rXq. Hence we can

identify AutpSq oSr and Autp rXq.

Finally, for each i “ 1, . . . , r, the natural isomorphism pri : Xi Ñ Xinduces the epimorphism

pri : Autp rXqXi Ñ AutpXq given by α ÞÑ pr´1i ˝α|Xi ˝ pri,

where ˝ denotes the usual composition of maps.

Suppose A ď Autp rXq with p|X|, |A|q “ 1. Let rΓ ď Autp rXq be such thatrΓ ď C

Autp rXqpAq. Suppose further that rΓA acts transitively on tX1, . . . , Xru.



Our objective is to control the structure of the groups A and rΓA. We needtwo preliminary results.

Lemma 5.19. Assume the notation in Notation 5.18. Write Bi “ pripAXiqfor each i P t1, . . . , ru.

(a) The groups Bi are cyclic.

(b) If rΓA acts transitively on tX1, . . . , Xru, then Bi and B1 are AutpXq-conjugate and the A-orbits on the set tX1, . . . , Xru all have the samelength.

Proof. Notice that Bi acts on X with p|X|, |Bi|q “ 1. By Theorem5.17, we have that Bi is cyclic. This proves part (a).

Now, since rΓA acts transitively on tX1, . . . , Xru, there is some αi P rΓAsuch that Xαi

1 “ Xi. It is easy to check that Bi “ prpAXiq “ prppAX1qαiq “

Bβi1 , where βi P AutpXq is given by βi ˝ pri “ pr1 ˝α

´1i |Xi .

Furthermore rΓ permutes transitively theA-orbits on tX1, . . . , Xru. Hencethese A-orbits all have the same length. This concludes the proof of part(b).

According to the classification of finite simple groups Schreier’s con-jecture holds, i.e., the outer automorphisms group OutpSq of every simplenon-abelian group S is solvable. In particular, if X is the universal coveringgroup of the non-abelian simple group S and π is the set of primes dividing|X|, then AutpXq is π-separable, and hence there are Hall π1-subgroups inAutpXq.

Using this fact one can determine a convenient group containing A.

Proposition 5.20. Assume the notation in Notation 5.18. Write Bi “pripAXiq for each i P t1, . . . , ru. Suppose that rΓA acts transitively ontX1, . . . , Xru. Let π be the set of prime divisors of |X|. Let H be a Hall π1-subgroup of AutpXq. Then A is AutpXqr-conjugate to a subgroup of H oSr.Also, Bi “ B1 for all i “ 1, . . . , r.

Proof. By the discussion preceding the statement of this propositionAutpXq is π-separable. Hence AutpXqrA is also π-separable. Let K “

AutpXqrAXSr. Notice that K is a π1-subgroup of Sr. Moreover, Hr ¸Sr

is a Hall π1-subgroup of AutpXqrA. Since A is a π1-subgroup of AutpXqrA,there exist a P A and α P AutpXqr, such that

Aaα “ Aα ď Hr ¸K ď H oSr.

For the latter part we may assume A ď H o Sr. Now, B1, Bi ď H areAutpXq-conjugate by Lemma 5.19(b). In particular |Bi| “ |B1|. Since H iscyclic, by Theorem 5.17, this implies Bi “ B1.

The next two propositions describe the structure of A and rΓA in thecase where A acts transitively on tX1, . . . , Xru.


90 5.4. Coprime action on simple groups

Proposition 5.21. Assume the notation in Notation 5.18. Suppose thatA ď H oSr. Write B “ prpAX1q. Then A is Hr-conjugate to a subgroup ofB oSr.

Proof. It is enough to prove the statement in the case where A actstransitively on tX1, ¨ ¨ ¨ , Xru by working on A-orbits.

If A ď H o Sr acts transitively on tX1, . . . , Xru, then for every i Pt1, . . . , ru, there exist ai P A such that

Xai1 “ Xi,

and hi P H such that for every x P X

px, 1, . . . , 1qai “ p1, . . . , xhi , . . . , 1q P Xi.

Let h “ p1H , h´12 , . . . , h´1

r q P Hr. We claim that Ah ď B oSr. First notice

that pr1ppAhqX1q “ pr1ppA

hX1qq “ B. Now, write pai “ h´1aih P A

h ď H oSr

for each i P t1, . . . , ru. Then

px, 1, . . . , 1qpai “ p1, . . . , x, . . . , 1q P Xi

for every x P X.

Let y P Ah. Then y “ py1, . . . , yrqρ P H oSr. Let i P t1, . . . , ru. Writej “ ρ´1piq, so that

Xyi “ Xj .

Since

px, 1 . . . , 1qpaiypa´1j “ pxyi , 1 . . . , 1q P X1,

for every x P X, we have that paiypa´1j P pAhqX1 . Consequently yj P B. This

argument applies for every i P t1, . . . , ru, hence the claim follows.

Recall the notation from the previous section: For H,K ď G, we denoteby Hm the direct product of m copies of H and we write ∆K ď Gm for

the diagonally embedded group K. We denote by q∆HK ď Gm the productof Hm and ∆K, whenever K ď NGpHq. If we want to emphasize thatq∆HK “ Hm∆K is constructed in Gm we write q∆m

HK.

Proposition 5.22. Assume the notation in Proposition 5.20. Let B “B1 and let Γ “ CAutpXqpBq. Suppose that A ď B o Sr and that A actstransitively on tX1, . . . , Xru. Then

rΓA ď pq∆BΓq ¸Sr.

Proof. Let c P rΓ. Then c “ pc1, . . . , crqρ with ci P AutpXq and ρ P Sr.Let a P AX1 . Then a “ pb1, . . . , brqσ with bi P B and σ P Sr. The equationac “ ca implies that

b1 “ c´11 bρ´1p1qc1 P B.

This holds for every a P AX1 . Hence c1 P NAutpXqpBq. By Theorem 5.17, wehave that NAutpXqpBq “ CAutpXqpBq “ Γ. Proceeding like this for elements

a P AXi , we conclude ci P Γ for every i P t1, . . . , ru. Hence rΓ ď Γ oSr.



Now, for a P A with a “ pb1, . . . , brqσ P B o Sr and c P rΓ with c “pc1, . . . , crqρ P Γ oSr the equation ac “ ca implies that

bicσ´1piq “ cibρ´1piq.

Hence cσ´1p1qc´1i P B for every i P t1, . . . , ru. Since A acts transitively on

tX1, . . . , Xru, we have thatcjc

´11 P B,

for every j P t1, . . . , ru. This proves that pc1, . . . , crq P Br∆Γ “ q∆BΓ. This

proves that rΓ ď pq∆BΓq ¸Sr.

We finally consider the case where rΓA acts transitively on tX1, . . . , Xru.

Proposition 5.23. Assume the notation in Proposition 5.20. Let B “

B1 and let Γ “ CAutpXqpBq. Suppose that A ď B oSr and that rΓA acts tran-sitively on tX1, . . . , Xru. Let m be the length of an A-orbit in tX1, . . . , Xru.Then for some τ P Sr we have that

Aτ ď pB oSmqrm and prΓAqτ ď ppq∆m

BΓq ¸Smq oS rm.

Proof. By Proposition 5.22 the statement holds when m “ r.

Let d “ rm. We may assume that the A-orbits on tX1, . . . , Xru are

exactly tX1, . . . , Xmu, . . . , tXpd´1qm`1, . . . , Xdmu after conjugating rΓA by

some τ P Sr. (Notice that Aτ and rΓτ satisfy the same hypotheses as A andrΓ.) This proves the first statement.

Let a “ pb1, . . . , brqσ P Aτ ď pB oSmq

d and c “ pc1, . . . , crqρ P rΓτ . Note

that σ P pSmqd, hence we can write σ “ σ1 ¨ ¨ ¨σd where σl P Sm permutes

the set tpl ´ 1qm` 1, . . . , lmu. The equation ac “ ca implies that

bicσ´1piq “ cibρ´1piq and σρ “ σ.

Notice that σρ “ σ implies that for every l P t1, . . . , du, we have that σρl “ σkfor a unique k P t1, . . . , du. Proceeding as in the first paragraph of the proofof Proposition 5.22 we can prove that ci P Γ. Also, arguing as in the secondparagraph of the proof of Proposition 5.22 we see that

cσ´1piqc´1i P B

for every i P t1, . . . , ru. We can proceed like this for every a P A. Since A istransitive on each tpl ´ 1qm` 1, . . . , lmu we conclude that

cjc´1l P B

for every j P tpl ´ 1qm ` 1, . . . , lmu and for every l P t1, . . . , du. Hence

pc1, . . . , crq P pBm∆Γqd “ pq∆m

BΓqd.Finally, since σρ “ σ for every σ coming from an element a P A, we

conclude that ρ permutes the set ttpl ´ 1qm ` 1, . . . , lmu | l “ 1, . . . , duand also permutes the elements of the set tpl ´ 1qm ` 1, . . . , lmu for eachl “ 1, . . . , d. Hence ρ P Sm o Sd. We conclude that c “ pc1, . . . , crqρ P

ppq∆mBΓq ¸Smq oSd.


92 5.5. The inductive Brauer-Glauberman condition

5.5. The inductive Brauer-Glauberman condition

We begin this section by defining the inductive Brauer-Glauberman condi-tion (see Definition 5.24 below). After that, we study consequences of thevalidity of the inductive Brauer-Glauberman condition for a simple non-abelian group S. The main result of this section is Theorem 5.27.

Definition 5.24. Let S be a non-abelian simple group and let X bethe universal covering group of S. We say that S satisfies the inductiveBrauer-Glauberman condition for the prime p if for every B ď AutpXqwith p|X|, |B|q “ 1 the following conditions are satisfied:

(i) For Z “ ZpXq, Γ “ CAutpXqpBq, C0 “ CXpBq and C “ C0Z, thereexists a Γ-equivariant bijection

ΩB : IBrBpXq Ñ IBrBpCq,

such that for every θ P IBrBpXq

pX ¸ Γθ, X, θq ąBr,c pC ¸ Γθ, C,ΩBpθqq.

(ii) For every positive integer m with p|X|,mq “ 1, there exists a fakem-th Galois action on IBrpC0q with respect to C0 ¸ Γ.

We can simplify a bit the verification of Definition 5.24 for all non-abeliansimple groups. First we only need to consider B up to AutpXq-conjugation.Moreover condition (ii) is true for every B if (ii) is true for B “ 1. Part ofthese simplifications is possible thanks to the fact that fake Galois actionsdo exist in p-solvable groups.

Remark 5.25. Let S be a non-abelian simple group and X the universalcovering group of S.

(a) The group S satisfies the inductive Brauer-Glauberman condition,if conditions (i) and (ii) in Definition 5.24 hold for some completeset of representatives of classes of AutpXq-conjugate subgroups Bof AutpXq with p|X|, |B|q “ 1.

(b) Let B ď AutpXq with p|X|, |B|q “ 1 and |B| ‰ 1 and assume thatCXpBq is quasi-simple. Then condition (ii) in Definition 5.24 holdsfor X and B, if for every integer m with p|X|,mq “ 1 there exists afake m-th Galois action on IBrpX1q with respect to X1 ¸AutpX1q,where X1 is the universal covering group of the unique non-abeliancomposition factor of CXpBq.

(c) Let B ď AutpXq with p|X|, |B|q “ 1 and |B| ‰ 1 and assume thatCXpBq is not quasi-simple. Then condition (ii) in Definition 5.24holds for X and B.

Proof. For the proof of (a) we suppose that (i) and (ii) from Definition5.24 are satisfied for the universal covering group X of some simple non-abelian group and for some B ď AutpXq with p|X|, |B|q “ 1. Assume thenotation Definition 5.24(i) with respect to B. Let α P AutpXq. DefineΩBαpχ

αq “ ΩBpχqα for every χ P IBrBpXq. Then ΩBα is a Γα-equivariant



bijection from IBrBαpXq onto IBrBαpCαq. By Lemma 4.22 we have that

condition (i) in Definition 5.24 is satisfied for ΩBα . Let m be an integerwith p|X|,mq “ 1. Let fm : IBrpC0q Ñ IBrpC0q give the fake m-th Galoisaction on IBrpC0q with respect to C0 ¸ Γ, as in Definition 4.37. Definef 1mpϕ

αq “ fmpϕqα for every ϕ P IBrpC0q. It is easy to prove an analogue of

Lemma 4.22 for m-th Galois conjugate modular character triples via Lemma4.35. This implies that f 1m gives a fake m-th Galois action on IBrpCα0 q withrespect to Cα0 ¸ Γα.

Now we prove parts (b) and (c). Let X be the universal covering groupof some non-abelian simple group S. Let B ď AutpXq with p|B|, |X|q “ 1.If B ‰ 1, then the group S is a simple group of Lie type and B is AutpXq-conjugate to some subgroup of the field automorphisms of X, see for exampleSection 2 of [MNS15]. By part (a) we may assume that B consists of fieldautomorphisms. By Theorem 2.2.7 of [GLS03], the group CXpBq is eitherquasi-simple, solvable or CXpBq P tB2p2q,G2p2q,

2 F4p2q,2 G2p3qu.

Suppose that CXpBq is not quasi-simple. Write C0 “ CXpBq. If C0 P

tB2p2q,G2p2q,2 F4p2q,

2 G2p3qu, then OutpC0q is cyclic and ZpC0q is trivial.Hence, it is easy to show that the identity yields a fake m-th Galois actionon IBrpC0q with respect to C0 ¸AutpC0q, for every positive integer m withp|X|,mq “ 1. If C0 is a solvable group, then Theorem 4.39 guarantees thatfor every m with p|X|,mq “ 1, there exists a fake m-th Galois action onIBrpC0q with respect to any G in which C0 is normal. This proves part (c).

In all other cases CXpBq is quasi-simple and there exists some non-abelian simple group S1 such that CXpBq is a central quotient of the univer-sal covering group X1 of S1. By assumption for every m with p|X1|,mq “ 1,there exists a fake m-th Galois action on IBrpX1q with respect to X1 ¸

AutpX1q. Since CAutpXqpBq ď AutpX1q this gives the required fake m-thGalois action on IBrpCXpBqq according to an analogue of Lemma 4.23 forfake Galois conjugate modular character triples. We use that if m is suchthat p|CXpBq|,mq “ 1, then also p|X1|,mq “ 1 by [Asc00, 33.12]. Thisproves (b).

Notation 5.26. Let S be a non-abelian simple group andX its universalcovering. We write Z “ ZpXq. If B ď AutpXq, then we write C0 “ CXpBq,C “ C0Z and Γ “ CAutpXqpBq. Recall Notation 5.13, and for a positive

integer r, write rX “ Xr, rZ “ Zr, rC “ Cr. Let ∆: X Ñ rX be the map

defined as in Notation 5.9. We also write qC “ q∆ZC0. For each i P t1, . . . , rulet Xi and pri be defined as in Notation 5.18.

Our aim in this section is to prove the following result and study someconsequences of it.

Theorem 5.27. Let S be a non-abelian simple group satisfying the induc-tive Brauer-Glauberman condition for the prime p. Let X be the universal

covering group of S, r a positive integer and A ď Autp rXq with p|A|, |X|q “ 1.



Write rΓ “ CAutp rXq

pAq. Suppose that rΓA acts transitively on the factors

tX1, . . . , Xru of rX. Then there exists a rΓ-equivariant bijection

rΩrX,A

: IBrAp rXq Ñ IBrApCrXpAq rZq,

such that for every χ P IBrAp rXq and χ1 “ rΩrX,Apχq

p rX ¸ rΓχ, rX,χq ąBr,c pCrXpAq rZ ¸ rΓχ,C

rXpAq rZ, χ1q.(5.2)

Remark 5.28. The above result illustrates why we need to require theexistence of fake Galois actions in Definition 5.24. Let X be the univer-sal covering group of a simple non-abelian group S. Let r be a positive

integer. Let A ď Sr act on rX by transitively permuting tX1, . . . , Xru.

Then CrXpAq “ ∆X ď rX. Let χ “ ϕ ˆ ¨ ¨ ¨ ˆ ϕ P IBrAp rXq. In par-

ticular Theorem 5.27 requires the existence of χ1 P IBrp∆Xq such that

p rX, rX,χq ąBr,c p∆X,∆X,χ1q. If we write χ1 “ ∆ϕ1 for ϕ1 P IBrpXq, then

ϕ1 must lie above λm, where λ P IBrpϕZpXqq. If one analyzes a little morethis example (namely the factor sets condition in this example), then it iseasy to see that ϕ1 needs to be a fake r-th Galois conjugate of ϕ.

We continue identifying Autp rXq “ AutpXq oSr as in Notation 5.18. For

A ď Autp rXq, write B “ pr1pAX1q ď AutpXq.

We prove Theorem 5.27 in a series of steps. We first prove a particularcase and after that we use this particular case to prove the general statement.

Now, we concentrate on proving Theorem 5.27 in the case where A actstransitively on tX1, . . . , Xru and A ď B oSr. Among other things, we wantto define a bijection

rΩrX,A

: IBrAp rXq Ñ IBrApCrXpAq rZq

such that corresponding characters give centrally isomorphic character triples.

We first determine the group CrXpAq rZ and some character sets with which

we will work.Recall Notation 5.26. If B ď AutpXq, then C0 “ CAutpXqpBq, C “ C0Z

and qC “ ∆ZC0.

Lemma 5.29. If A acts transitively on tX1, . . . , Xru and A ď B o Sr,then

(a) CrXpAq “ ∆C0 “ C

rXpB oSrq. In particular, C

rXpAq rZ “ qC,

(b) IBrAp rXq “ tθ ˆ ¨ ¨ ¨ ˆ θ | θ P IBrBpXqu,

(c) IBrAp rZq “ tν ˆ ¨ ¨ ¨ ˆ ν | ν P IBrBpZqu,(d) IBrBpCq “ tϕ ¨ ν | ϕ P IBrpC0q and ν P IBrBpZqu,

(e) IBrAp rCq “ tpϕ ˆ ¨ ¨ ¨ ˆ ϕq ¨ pν ˆ ¨ ¨ ¨ ˆ νq | ϕ P IBrpC0q and ν PIBrBpZqu,

(f) IBrAp qCq “ tp∆ϕq ¨ µ | ϕ P IBrpC0q and µ P IBrAp rZqu.



Proof. The equalities CrXpAq “ ∆C0 “ C

rXpB oSrq easily follow from

Lemma 2.2 of [IN96]. Then qC “ q∆ZC0 “ CrXpAq rZ, as wanted. The rest

easily follow from the definitions (use Lemma 4.27 for parts (d), (e) and(f)).

In the following proposition, we introduce a key bijection rfr : IBrAp rCq Ñ

IBrAp qCq that will help us to define the map rΩrX,A

.

Proposition 5.30. In the situation of Theorem 5.27, suppose that A

acts transitively on tX1, . . . , Xru and A ď B oSr. Let qΓ “ q∆BΓ and qY “

qC ¸ pqΓ¸Srq. Then there exists a ∆Γ-equivariant bijection

rfr : IBrAp rCq Ñ IBrAp qCq

such that for every rψ P IBrAp rCq and qψ “ rfrp rψq we have

p rC qYrψ, rC, rψq ąBr,c pqY

rψ, qC, qψq.

Proof. Write Z0 “ ZpC0q, rZ0 “ Zr0 and rC0 “ Cr0 . Note that theassumption that A acts transitively on tX1, . . . , Xru with p|X|, |A|q “ 1implies pr, |X|q “ 1. Since S satisfies the inductive Brauer-Glaubermancondition, there exists a fake Galois r-th action on IBrpC0q with respect toC0 ¸ Γ. Let fr give a fake r-th Galois action as in Definition 4.37.

Let rψ P IBrAp rCq. Recall rC “ rC0rZ is the central product of rC0 and rZ.

By Theorem 5.17, rC0 X rZ “ rZ0. By Lemma 5.29(f), we have that rψ “ rϕ ¨ rν

for some rϕ “ ϕˆ ¨ ¨ ¨ ˆ ϕ P IBrAp rC0q and rν “ ν ˆ ¨ ¨ ¨ ˆ ν P IBrAp rZq whereϕ and ν lie over the same λ P IBrpZ0q. We define

rfrp rψq “ ∆pfrpϕqq ¨ rν.

By Lemma 5.29, we have that rfr is a bijection. Note that rfr is ∆Γ-

equivariant as fr is Γ-equivariant. In particular, for qψ “ rfrp rψq we have

that qYrψ“ qY

qψ.

Let ϕ1 “ frpϕq and write Y0 “ C0 ¸ Γϕ. Since fr yields a fake r-thGalois action on IBrpC0q with respect to Y0, we have that

pY0, C0, ϕqprq « pY0, C0, ϕ

1q.(5.3)

Let qC0 “ q∆Z0C0. Write rλ “ λ ˆ ¨ ¨ ¨ ˆ λ P IBrp rZ0q, where IBrpϕZ0q “

tλu “ IBrpνZ0q, and qϕ “ ∆ϕ1 ¨ rλ P IBrp qC0q. Write also qY0 “ q∆Z0BY0. SinceZ0B ď CY0pC0q, by Corollary 5.15 we have that Equation (5.3) yields

p rC0pqY0 ¸Srq, rC0, rϕq ąBr,c pqY0 ¸Sr, qC0, qϕq.(5.4)

We have that qC “ rZ qC0. Then Equation (5.4) together with Lemma 4.28imply

p rC qYrψ, rC, rψq ąBr,c pqYψ,

qC, qψq.



Proposition 5.31. If A acts transitively on tX1, . . . , Xru and A ď B oSr, then Theorem 5.27 holds.

Proof. Let ΩB be the map given by Definition 5.24(i). For θ P IBrBpXq,

we write θ1 “ ΩBpθq. We define a map rΩ “ rΩrX,A

by

rΩ: IBrAp rXq Ñ IBrBp qCq

θ ˆ ¨ ¨ ¨ ˆ θ ÞÑ rfrpθ1 ˆ ¨ ¨ ¨ ˆ θ1q

where rfr is given by Proposition 5.30. By Lemma 5.29, this map is well-

defined. In fact, since rfr is bijective, then rΩ is also bijective (see Lemma5.29(f)).

Recall rΓ “ CAutp rXq

pAq. Write Υ “ q∆BΓ ¸ Sr “ ∆ΓpB o Srq. By

Proposition 5.22rΓ ď rΓA ď Υ.

In order to prove that rΩ is rΓ-equivariant we show that rΩ is actually Υ-

equivariant. In view of the description of IBrAp rXq and IBrAp rCq given inLemma 5.29, it follows that B o Sr acts trivially on these sets. Hence our

bijection rΩ is B oSr-equivariant. By definition, ΩB is Γ-equivariant and by

Proposition 5.30, rfr is ∆Γ-equivariant. Hence rΩ is also ∆Γ-equivariant (soΥ-equivariant).

It remains to prove that for every χ P IBrAp rXq we have that

p rX ¸ rΓχ, rX,χq ąBr,c p rC ¸ rΓχ, rC, rΩpχqq.

Let χ P IBrAp rXq and θ P IBrBpXq with χ “ θ ˆ ¨ ¨ ¨ ˆ θ. By Definition5.24(i), we have that θ and θ1 “ ΩBpθq satisfy

pX ¸ Γθ, X, θq ąBr,c pC ¸ Γθ, C, θ1q.

Let rψ “ θ1 ˆ ¨ ¨ ¨ ˆ θ1 P IBrp rCq. The equation above together with Corollary4.25 imply that

p rX ¸ pΓθ oSrq, rX,χq ąBr,c p rC ¸ pΓθ oSrq, rC, rψq.

Using Υ ď Γ oSr and Υχ ď Γθ oSr we deduce that

p rX ¸Υχ, rX,χq ąBr,c p rC ¸Υχ, rC, rψq.(5.5)

Let qY “ qC ¸ Υ and qψ “ rfrp rψq. Then qψ “ rΩpχq. By Proposition 5.30 weknow that

p rC qYrψ, rC, rψq ąBr,c pqY

rψ, qC, qψq.

Because of rC ¸Υχ “ rC qYrψ

and qC ¸Υqψ“ qY

qψ, the equation above is exactly

p rC ¸Υχ, rC, rψq ąBr,c p qC ¸Υrψ, qC, qψq.(5.6)



Since ąBr,c is transitive, Equations (5.5) and (5.6) imply that

p rX ¸Υχ, rX,χq ąBr,c p qC ¸Υrψ, qC, qψq.

Since rΓχ ď Υχ “ Υrψ

and qψ “ rΩpχq this proves the statement.

Remark 5.32. Assume the situation described in Proposition 5.31 aswell as the notation of Theorem 5.27. The proof of Proposition 5.31 actually

shows that the conclusions of Theorem 5.27 also hold with Υ “ q∆BΓ¸Sr ĚrΓ in place of rΓ.

Our next step is to prove Theorem 5.27 in the case where A no longer actstransitively on tX1, . . . , Xru but the structure of A is somehow controlled,

namely A ď pB oSrqrm for some divisor m of r.

Proposition 5.33. Let m be the length of some A-orbit on tX1, . . . , Xru.

If rΓA acts transitively on tX1, . . . , Xru and A ď pB oSmqrm, then Theorem

5.27 holds.

Proof. Let Λ “ tX1, . . . , Xru and Λ1, . . . ,Λd be the A-orbits on Λ.

Notice that rΓ permutes the A-orbits transitively so that d “ rm. The

assumption A ď pB o Smqrm implies that Λj “ tXpj´1qm`1, . . . , Xjmu for

every 1 ď j ď d. By Proposition 5.23 we have that

ACAutp rXq

pAq “ ArΓ ď Υ,

where Υ “ pq∆mBΓ ¸ Smq o Sd (because in this case we can take τ “ 1 in

Proposition 5.23).

For every j P t1, . . . , du, let

XΛj “ź

Y PΛj

Y and ZΛj “ź

Y PΛj

ZpY q.

Clearly A acts on XΛj with p|A|, |XΛj |q “ 1. Let Υj be the projection ofStabΥpXΛj q into AutpXΛj q. Then, it is easy to show that Υj is isomorphic

to q∆mBΓ ¸ Sm. By Lemma 5.31 (and using Remark 5.32), there is a Υ1-

equivariant bijection

rΩΛ1,A : IBrApXΛ1q Ñ IBrAp qCΛ1q,

where qCΛj “ CXΛjpAqZΛ1 . Furthermore for every χ1 P IBrApXΛ1q and

qχ1 “ rΩΛ1,Apχ1q we have

pXΛ1 ¸ pΥ1qχ1 , XΛ1 , χ1q ąBr,c p qCΛ1 ¸ pΥ1qχ1 ,qCΛ1 , qχ1q.(5.7)

For j P t2, . . . , du, we define Υj-equivariant bijections

rΩΛj ,A : IBrApXΛj q Ñ IBrAp qCΛj q



from rΩXΛ1,A via the permutation action of Sd on tXΛ1 , . . . , XΛdu. For every

χj P IBrApXΛj q and qχj “ rΩΛj ,Apχjq we have

pXΛj ¸ pΥjqχj , XΛj , χjq ąBr,c p qCΛj ¸ pΥjqχj ,qCΛj , qχjq(5.8)

by a transfer of Equation (5.7) via Lemma 4.22.

Note that rX “ XΛ1 ˆ ¨ ¨ ¨ ˆ XΛd is an internal direct product and

A stabilizes each XΛj . Hence IBrAp rXq is in natural correspondence with

IBrApXΛ1qˆ ¨ ¨ ¨ˆ IBrApXΛdq. Analogously CrXpAq rZ “ qC “ qCΛ1ˆ¨ ¨ ¨ˆ

qCΛd

is an internal direct product, so that IBrApCrXpAqq “ IBrAp qCΛ1q ˆ ¨ ¨ ¨ ˆ

IBrAp qCΛdq.

Define rΩrX,A

: IBrAp rXq Ñ IBrp qCq by

χ1 ˆ ¨ ¨ ¨ ˆ χd ÞÑ rΩΛ1,Apχ1q ˆ ¨ ¨ ¨ ˆ rΩΛd,Apχdq.

We see that rΩrX,A

is a well-defined pΥ1ˆ¨ ¨ ¨ˆΥdq-equivariant bijection. By

definition rΩrX,A

is Sd-equivariant, where Sd is identified with the subgroup

of Υ acting on the groups XΛi by permutation. Hence rΩrX,A

is Υ-equivariant.

It is easy to prove that every character in IBrAp rXq is pΥ1 ˆ ¨ ¨ ¨ ˆ Υdq-conjugate to some χ “ χ1ˆ¨ ¨ ¨ˆχd where either χi and χj are Sd-conjugateor χi and χj are not Υ-conjugate (again Sd is identified with the subgroupof Υ acting on the XΛj groups by permutation). In particular, the stabilizerΥχ of χ in Υ satisfies

Υχ “ ppΥ1qχ1 ˆ ¨ ¨ ¨ pΥdqχdq ¸ pSdqχ.

The Equation (5.8) for each j P t1, . . . , du together with Corollary 4.25 implythat the character χ satisfies

p rX ¸Υχ, rX,χq ąBr,c p qC ¸Υχ, qC,χ1q,

where χ1 “ rΩpχq “ qχ1 ˆ ¨ ¨ ¨ ˆ qχd. Of course, since rΓ ď Υ, we deduce

p rX ¸ rΓχ, rX,χq ąBr,c p qC ¸ rΓχ, qC,χ1q.

By Lemma 4.22 this finishes the proof.

Proof of Theorem 5.27 . By Proposition 5.20 there exists some α PAutpXqr such that Aα ď B o Sr, where B is the projection of AX1 on

AutpXq. Let m be the length of an A-orbit on tX1, . . . , Xru. Since rΓ actstransitively on the A-orbits, d “ rm is the number of A-orbits. Let τ P Sd

be as given in Proposition 5.22. Then Aατ ď pB o Smqd. Let rΩ

rX,Aατbe

the rΓατ -equivariant bijection given by Proposition 5.33. Define rΩrX,A

by

χ ÞÑ rΩrX,Aατ

pχατ q for every χ P IBrAp rXq. It is easy to check that rΩrX,A

is

a rΓ-equivariant bijection. Use Lemma 4.22 to check the central character

triple isomorphism condition with respect to rΩrX,A

.



The importance of Theorem 5.27 for us is illustrated in the followingtwo results, which are consequences of it.

Theorem 5.34. Suppose that A acts coprimely on a finite group G. LetK Ÿ G be an A-invariant perfect subgroup. Suppose that G “ KCGpAq.Write C “ CGpAq and M “ K X C. Suppose further that A acts triviallyon N “ ZpGq ď K, KN “ S1 ˆ ¨ ¨ ¨ ˆ Sr – Sr, where S is a non-abeliansimple group, and CA permutes transitively tS1, . . . , Sru. If S satisfies theinductive Brauer-Glauberman condition, then there exists a C-equivariantbijection

Ω1 : IBrApKq Ñ IBrpMq,

such that

pGχ,K, χq ąBr,c pCχ,M, χ1q

for every χ P IBrApKq and χ1 “ Ω1pχq.

Proof. Since G “ KCGpAq, then CApKq “ CApGq and we may as-

sume that A acts faithfully on K, i.e., A ď AutpKq. Let rS “ Sr. Let X be

the universal covering group of S. Then, rX “ Xr is the universal covering

group of rS. Write rZ “ Zp rXq. Since K is a covering of rS, there exists an

epimorphism ε : rX Ñ K with L “ kerpεq ď rZ. In fact, rX is the universalcovering of K.

The map ε induces an isomorphism Autp rXqL Ñ AutpKq. Hence, thegroups A and C “ CCGpKqCGpKq can be seen as groups of automorphisms

of rX. In fact, under this identification, the group AC ď ACAutp rXq

pAq acts

transitively on the factors of rX. Since p|A|, |K|q “ 1, we have p|A|, | rX|q “ 1

by [Asc00, 33.12]. Write rΓ “ CAutp rXq

pAq and qC “ CrXpAq rZ. By Theorem

5.27, there exists a rΓ-equivariant bijection

rΩ “ rΩrX,A

: IBrAp rXq Ñ IBrAp qCq

such that

p rX ¸ rΓχ, rX,χq ąBr,c p qC ¸ rΓχ, qC, rΩpχqq

for every χ P IBrAp rXq. Since C ď rΓ, we have that rΩ is C-equivariant and

p rX ¸ Cχ, rX,χq ąBr,c p qC ¸ Cχ, qC,χ1q(5.9)

for every χ P IBrAp rXq and χ1 “ rΩpχq. By Definition 4.19(ii), we have that

for every χ P IBrAp rXq, the characters χ and χ1 “ rΩpχq lie over the same

character λ P rZ. We deduce easily that

rΩpIBrAp rX | 1Lqq “ IBrAp qC | 1Lq.

Note that εp qCq “ εpCrXpAq rZq “ CKpAq “ M and IBrApMq “ IBrpMq.

Hence rΩ defines, via ε, a C-equivariant bijection

Ω1 : IBrApKq Ñ IBrpMq.


100 5.6. A reduction theorem

Notice that CrX¸C

p rXq “ rZ, εp rZLq “ N “ ZpKq “ CK¸CpKq and L ď

kerpχq X kerpχ1q for every χ P IBrAp rXLq and χ1 “ rΩpχq P IBrAp qCLq. ByLemma 4.23, Equation (5.9) implies that

pK ¸ Cχ,K, χq ąBr,c pM ¸ Cχ,M, χ1q

for every χ P IBrApKq and χ1 “ Ω1pχq. Finally, a direct application ofTheorem 4.26 yields

pGχ,K, χq ąBr,c pCχ,M, χ1q.

Corollary 5.35. Suppose A that acts coprimely on G. Let KŸG be A-invariant. Suppose that G “ KCGpAq. Write C “ CGpAq and M “ KXC.Suppose further that A acts trivially on N “ ZpGq ď K, KN “ S1 ˆ

¨ ¨ ¨ ˆ Sr – Sr, where S is a non-abelian simple group, and CA permutestransitively tS1, . . . , Sru. If S satisfies the inductive Brauer-Glaubermancondition for the prime p, then there exists a bijection


such that

pGχ,K, χq ąBr,c pCχ,M, χ1q

for every χ P IBrApKq and χ1 “ Ω1pχq.

Proof. Let K1 “ rK,Ks. Since KN is a direct product of simplenon-abelian groups, it follows that K1 is perfect and K “ K1N . Let N1 “

N XK1. Notice that CGpK1q “ CGpKq. Also K is the central product ofK1 and N . Write M1 “M XK1.

Let Ω11 : IBrApK1q Ñ IBrpM1q be the bijection given by Theorem 5.34.Every χ P IBrApKq has the form χ1 ¨ µ, where χ1 P IBrApK1q, µ P IBrpNqand both characters lie over the same Brauer character of N1. Define

Ω1 : IBrApKq Ñ IBrpMq

χ1 ¨ µ ÞÑ Ω11pχ1q ¨ µ.

It is clear that Ω1 is a C-equivariant bijection. Let χ “ χ1 ¨ µ P IBrApKq.By Theorem 5.34 we have pGχ1 ,K1, χ1q ąBr,c pCχ1 ,M1,Ω

11pχ1qq. A direct

application of Lemma 4.28 implies

pGχ,K, χq ąBr,c pCχ,M,Ω1pχqq.

5.6. A reduction theorem

We are finally ready to prove Theorem G. Since we need to use a stronginductive argument, we first prove a relative to normal subgroups version ofTheorem G below.

Theorem 5.36. Let A act coprimely on G. Let N Ÿ G be stabilized byA and write C “ CGpAq. Let θ P IBrApNq. Suppose that the non-abelian



simple groups involved in GN satisfy the inductive Brauer-Glauberman con-dition. Then

|IBrApG|θq| “ |IBrpCN |θq|.

Proof. We proceed by induction on |G : N |.

Step 1. We may assume θ is G-invariant.Let T “ Gθ. Then CN X T “ pCNqθ. By the Clifford correspondence forBrauer characters (see Theorem 4.7) we have that

|IBrApG|θq| “ |IBrApT |θq| and |IBrpCN |θq| “ |IBrpCN X T |θq|.

If T ă G, then by induction hypothesis with respect to |T : N | ă |G : N |,we have that

|IBrApT |θq| “ |IBrpCT pAqN |θq| “ |IBrpCN X T |θq|.

Step 2. We may assume that N ď ZpGAq is a p1-group.By Theorem 8.28 of [Nav98], there exists a strong isomorphism of modularcharacter triples

pσ, τq : pGA,N, θq Ñ pΓ,M, ϕq

such that M ď ZpΓq is a p1-group. Whenever N ď H ď GA, we writeHτ to denote the subgroup of Γ such that τpHNq “ Hτ M . Then A –

τpANNq “ pANqτ M , so that pANqτ M acts on Gτ M as A acts on GNand pANqτ M acts trivially on M . Therefore

pCNqτ M “ τpCNNq “ CGτ M ppANqτ Mq “ CGτ ppANq

τ qM.

By Theorem 4.12(a), θ extends to AN , and hence ϕ extends to pANqτ .Recall that ϕ is a linear character since M ď ZpΓq. Let π be the setof primes dividing |A|. Write ϕ “ ϕπϕπ1 . Recall that ϕπ and ϕπ1 , theπ-part and π1-part of ϕ, are powers of ϕ. In particular, ϕπ extends topANqτ . If q R π, then by Theorem 4.12(b) we have that ϕπ extends to Qfor every QM P SylqpΓMq. Thus ϕπ extends to Γ by Theorem 4.11. Byparts pdq and pbq of Lemma 4.17, the modular character triple pΓ,M, ϕq isstrongly isomorphic to pΓ,M, ϕπ1q and we may assume that ϕπ1 is faithful.Write ϕ1 “ ϕπ1 . We have that |M | “ opϕ1q is a π1-number. Hence Mhas a complement B in pANqτ by Schur-Zassenhaus’ theorem [Isa08, Thm.3.5]. Thus B acts coprimely on Gτ and pCNqτ M “ CGτ pBqM . SincepGA,N, θq is strongly isomorphic to pΓ,M, ϕ1q we have that

|IBrApG|θq| “ |IBrBpGτ |ϕ1q| and |IBrpCN |θq| “ |IBrpCGτ pBq|ϕ

1q|.

Moreover, since GN – Gτ M , then the simple groups involved in Gτ Msatisfy the inductive Brauer-Glauberman condition for the prime p. Hence,the claim follows if it follows for Gτ , M , B and ϕ.

Step 3. We may assume G “ KC for every A-invariant K with N ă

K Ÿ G.


102 5.6. A reduction theorem

Let N ă KŸG be A-invariant. We have that C acts on IBrApK|θq. Let B bea complete set of representatives of C-orbits on IBrApK|θq. Let K ď H ď Gand ψ P IBrApH|θq. By Theorem 4.6, we have that HK acts transitively onIBrpψKq. Also A acts on IBrpψKq. Since p|A|, |HK|q “ 1, by Glauberman’sLemma [Isa76, Lem. 13.8] and [Isa76, Cor. 13.9] there is some A-invariantcharacter in IBrpψKq and any two of them are C-conjugate. This provesthat every ψ P IBrApH|θq lies over a unique element of B. By the previousargument for H “ G and H “ CK we have that

|IBrApG|θq| “ÿ

ηPB|IBrApG|ηq| and |IBrApCK|θq| “

ÿ

ηPB|IBrApCK|ηq|.

By the inductive hypothesis |IBrApG|ηq| “ |IBrpCK|ηq| for every η P B.Since A acts coprimely on CKK and CCKKpAq “ CKK, we have thatIBrApCK|ηq “ IBrpCK|ηq by Lemma 5.3. Hence |IBrApG|θq| “ |IBrApCK|θq|.If CK ă G, then by induction |IBrApCK|θq| “ |IBrpC|θq|, and the claimfollows.

Step 4. We may assume OppGq “ 1.Write O “ OppGq. If O ą 1, then |GO : NOO| ă |G : N |. By Lemma 4.4,O ď kerpϕq for every ϕ P IBrpGq. To prove the claim use that CGOpAq “COO by coprime action and the inductive hypothesis.

Step 5. Every chief factor KN of GA with K ď G is a direct productof isomorphic non-abelian simple groups and N “ ZpGq.Let KN be a chief factor of GA with K ď G. We may assume that G “ KCby Step 3 and that KN is not a p-group by Step 4. If KN is a p1-group,then |IBrApG|θq| “ |IBrpC|θq| by Corollary 5.8. Hence we can assume thatGA has no abelian chief factor of the form KN with K ď G. In particularN “ ZpGq.

Final Step. Let KN be a chief factor of GA with K ď G. By Step 4 wemay assume G “ KC. By Step 5 we have that KN – S1ˆ ¨ ¨ ¨ ˆSr, wherethe Si are simple non-abelian groups. Notice that CA permutes transitivelythe groups Si in KN and hence they are all isomorphic. Since S “ S1 is in-volved in GN , then S satisfies the inductive Brauer-Glauberman condition.Write M “ C XK. By Corollary 5.35 there is a C-equivariant bijection


such that pGη,K, ηq ąBr,c pCη,M,Ω1pηqq for every η P IBrApKq. We writeη1 “ Ω1pηq for every η P IBrApKq. Since N ď CGpKq, then Ω1 actuallyyields a bijection IBrApK|θq Ñ IBrpM |θq. Let B be a set of representativesof C-orbits on IBrApK|θq. Every element of IBrApG|θq lies over a uniqueelement of B as in Step 3. Since Ω1 is C-equivariant, we have that the settη1 | η P Bu is a complete set of representatives of C-orbits on IBrpM |θq.



Hence

|IBrApG|θq| “ÿ

ηPB|IBrApG|ηq| and |IBrpC|θq| “

ÿ

ηPB|IBrpC|η1q|.

For every η P B, we have IBrApG|ηq “ IBrpG|ηq by Lemma 5.3 and

pGη,K, ηq ąBr,c pCη,M, η1q

by Corollary 5.35. In particular, |IBrpGη|ηq| “ |IBrpCη|η1q| for every η P B.

The result follows then by using Theorem 4.7.

Corollary 5.37. Let A be a group that acts coprimely on a group G.Suppose that every simple non-abelian group involved in G satisfies the in-ductive Brauer-Glauberman condition with respect to the prime p. Then theactions of A on the irreducible Brauer characters of G and on the p-regularclasses of G are permutation isomorphic

Proof. For every B ď A, Theorem 5.36 with N “ 1 and B playingthe role of A guarantees that |IBrBpGq| “ |IBrpCGpBqq|. The map K ÞÑ

K X CGpBq is a well-defined bijection between the set of B-invariant p-regular classes of G and the set of p-regular conjugacy classes of CGpBq.Hence the number of B-invariant irreducible Brauer characters of G equalsthe number of B-invariant p-regular conjugacy classes of G. By Lemma13.23 of [Isa76], this proves the statement.

5.7. Some examples

In this final section, we try to answer some questions naturally related toour topic. Recall that if A acts coprimely on G and p is a prime, then weare studying if

|IBrApGq| “ |IBrpCq|,

where C “ CGpAq (with respect to p-Brauer characters). If G is p-solvable,then we have already mentioned that K. Uno [Uno83] proved the aboveequality. In fact, he established a canonical bijection

˚ : IBrApGq Ñ IBrpCGpAqq

which behaves exactly as the Glauberman correspondence. In particular, ifA is a q-group for some prime q and ϕ P IBrApGq, then

ϕC “ eϕ˚ ` q∆,

where q does not divide e, and ∆ is zero or a Brauer character of C. Itis natural to ask if the equation above holds without restrictions on thestructure of G. Unfortunately, the next example shows that the answer isnegative.

We let G be the simple group 2B2p2q of order 26 ¨ 5 ¨ 7 ¨ 13. It is wellknown that G has an automorphism σ of order 3. Call A “ xσy. Then

C “ CGpAq “ C5 : C4


104 5.7. Some examples

is a Frobenius group of order 22 ¨5. Let us set p “ 13. The irreducible charac-ters of G can be written as tχ1, α14, β14, α35, β35, γ35, χ64, α65, β65, γ65, χ91u,where the subscript in each case denotes the degree of the character. Fur-thermore, every irreducible character lifts an ordinary p-Brauer character,except the character of degree 64. We can write

IBrpGq “ tϕ1, ϕ2, ϕ3, ϕ4, ϕ5, ϕ6, ϕ7, ϕ8u,

where ϕ1 “ pχ1q0, ϕ2 “ pα14q

0, ϕ3 “ pβ14q0, ϕ4 “ pα35q

0 “ pβ35q0 “ pγ35q

0,ϕ5 “ pα65q

0, ϕ6 “ pβ65q0, ϕ7 “ pγ65q

0, ϕ8 “ pχ91q0. Also

pχ64q0 “ ϕ1 ` ϕ2 ` ϕ3 ` ϕ4.

We have checked that

IBrApGq “ tϕ1, ϕ2, ϕ3, ϕ4, ϕ8u.

The centralizer C is a p1-group, and IrrpCq “ t1, λ, ε, ε, δu, where ε is alinear character of order 4, λ “ ε2 and δp1q “ 4. We have computed therestrictions of the A-invariant Brauer characters to C.

pϕ1qC “ 1C0

pϕ2qC “ 2ε` 3δ

pϕ3qC “ 2ε` 3δ

pϕ4qC “ 2 ¨ 1C0 ` 3λ` ε` ε` 7δ

pϕ8qC “ 3 ¨ 1C0 ` 4λ` 6ε` 6ε` 18δ .

Hence, we see that all the restrictions have a unique irreducible constituentwith multiplicity not divisible by 3, except for ϕ4 whose restriction has 4irreducible constituents with multiplicity not divisible by 3. This fact is notsurprising in this case where C is a p1-group, since every ϕ P IBrApGq differ-ent from ϕ4 lifts to a unique χ P IrrpGq, that is, therefore, A-invariant. HenceϕC “ χC has the desired decomposition by Glauberman’s correspondence.Also, we see that ϕ4 does not lift to an A-invariant ordinary irreduciblecharacter. This raises a natural question on the Glauberman-Isaacs corre-spondence: if χ P IrrApGq lifts ϕ P IBrpGq, is it true that the Glauberman-Isaacs correspondent χ˚ P IrrpCGpAqq lifts an irreducible Brauer characterof CGpAq? This is the case for p-solvable groups, as shown in [SG94].

We come back to our example. If we were asked to establish a naturalbijection between IBrApGq and IBrpCq, we would give

ϕ1 ÞÑ 1C0

ϕ2 ÞÑ ε

ϕ3 ÞÑ ε

ϕ4 ÞÑ δ

ϕ8 ÞÑ λ .



As a curiosity, we remark that the fields of values between character corre-spondents under this bijection are the same.

Qpϕ2q “ Qpiq “ QpεqQpϕ3q “ Qpiq “ QpεqQpϕ4q “ Q “ QpδqQpϕ8q “ Q “ Qpλq .

It is not totally reasonable to expect that something like this is going tohappen in general, since as we know by now GalpQ|G|Qq does not act onIBrpGq. (Moreover, for every ϕ P IBrApGq, we see that there exists someg P C0 such that opgq “ fϕ. In the ordinary case, since the Glauberman-Isaacs correspondence preserves fields of values, Feit’s conjecture impliesthat if χ P IrrApGq, then there exists some c P CGpAq such that opcq “ fχ.)

We come back for a moment to ordinary character theory. Let q be aprime. It is well-known that the proof of the McKay conjecture in the q-solvable case heavily depends on the Glauberman correspondence. In fact,in groups with a normal q-complement, the McKay conjecture is essentiallythe Glauberman count. Namely, if G “ K ¸ Q, where Q P SylqpGq, thenN “ NGpQq “ CKpQq ˆQ, by elementary group theory. Therefore

|Irrq1pNq| “ |IrrpCKpQqq||QQ1|.

Also, every χ P Irrq1pGq restricts to θ “ χK P IrrQpKq (by Theorem 1.10).

Conversely, every θ P IrrKpQq extends canonically to θ P Irrq1pGq (by Theo-rem 1.13). Hence, by Gallagher’s theory (see Theorem 1.12, we have that

Irrq1pGq “ tβθ | for θ P IrrQpKq and β P IrrpQQ1qu,

where we identify the characters of QQ1 with the characters of GKQ1) itfollows that

|Irrq1pGq| “ |Q : Q1||IrrKpQq|.

Therefore, the McKay conjecture in this case reduces to the Glaubermancount |IrrQpKq| “ |IrrpCKpQqq|.

Now, we slightly change the notation that we have used throughout thischapter. Recall that we have been studying whether or not the equality

|IBrQpKq| “ |IBrpCKpQqq|

holds whenever a group Q acts coprimely on K (with respect to p-Brauercharacters, where p is any prime). Suppose that Q is a q-group and letG “ K ¸Q. In view of the previous discussion it makes sense to ask if

|IBrq1pGq| “ |IBrq1pNGpQqq|.

(Notice that this is a version of McKay for (p)-Brauer characters in thesimplest case in which the group has a normal q-complement.) The answer


106 5.7. Some examples

is negative, as pointed out to us by P. H. Tiep. If K “ SL3p32q, q “ 5,Q “ C5, G “ K ¸Q and p “ 31, then

|IBrq1pGq| ‰ |IBrq1pNGpQqq|.

Since Q is cyclic, in this case, we do have that |IBrQpKq| “ |IBrpCKpQqq|.The problem is that degrees of irreducible Brauer characters do not dividethe order of the group. In some sense, this example explains why the McKayand the coprime counting conjectures, need separate reductions. Althoughclosely related, there does not seem a simultaneous generalization of bothof them.


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